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CHAPTER 3

Heat transfer to objects in pool fires

J.P. Spinti, J.N. Thornock, E.G. Eddings, P.J. Smith & A.F. SarofimDepartment of Chemical Engineering, University of Utah, USA.

Abstract

In accident scenarios involving fire and the transport of explosive material, the time available for escape is dependent on the heat transfer rate from the fire to the energetic material. A review is pre-sented of historical modeling approaches that draw on empiricism for estimating both heat flux from fires and fire hazard. While such methods can be used for conservative estimates of heat flux in deter-mining safe separation distances, they cannot be used in situations where overestimating the heat flux may underestimate the hazard, such as the heating of high-energy explosives. Next, a large eddy simulation (LES) technique for addressing fire phenomena with embedded, heat sensitive objects is described. With the advent of high performance computing, LES is emerging as a powerful tool for resolving a large set of spatial and temporal scales in fires and for capturing observed pool fire phenomena such as visible flame structures. The development of the LES approach described here is based on verification and validation (V&V) principles, utilizing a V&V hierarchy that is focused on the intended use of the simulation. This LES approach couples surrogate fuel representations of complex hydrocarbon fuels, reaction models for incorporation of the detailed chemical kinet-ics associated with the surrogate fuel, soot formation models, models for unresolved turbulence/chemistry interactions, radiative heat transfer models, and modifications to the LES algorithm for computing heat transfer to objects. The chapter concludes with an analysis of simulation and experi-mental data of heat transfer to embedded objects in large JP-8 pool fires and of time to ignition of an energetic device in such a fire. The analysis considers the role of validation, sensitivity analysis and uncertainty quantification in moving toward predictivity.

1 Introduction

Explosives are transported via highway, rail line, and air for use in mining, space exploration, building demolition, pyrotechnics, avalanche control, and military applications. In addition, cer-tain hydrocarbons, most notably liquefied petroleum gas (LPG, mainly composed of propane), can explode when the storage vessel is heated by an external fire, resulting in the so-called boil-ing liquid expanding vapor explosion (BLEVE). For these events, the time to explosion is critical as it determines the time available for first responders to intervene and for those at the scene of an accident to escape.

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70 Transport Phenomena in Fires

On August 10, 2005, a semi-trailer truck carrying 38,000 pounds of mining explosives tipped over, skidded across the pavement, caught fire, and then detonated in Spanish Fork Canyon, Utah. The driver was negotiating a sharp turn at an excessive speed when the accident occurred. Eyewitnesses estimated a time of three minutes from the start of the fire to the detonation event. The blast left a crater 30 feet deep and 70 feet wide in the road, and the truck was reduced to shards of metal, frayed pieces of tire, and an engine block. This incident and others like it provide the motivation for calculat-ing the potential hazard of an explosive device immersed in a pool fire of transportation fuel.

Heat transfer to objects in or near pool fires has been the subject of study for decades. Tradition-ally, the focus has been on determining a safe separation distance from the fire. Calculation with a conservatively high heat flux provided a good margin of safety. However, there are times when conservative estimations of heat flux are inadequate for determining the magnitude of the hazard, particularly when dealing with containers of energetic materials. For example, some energetic materials may detonate under slow heating (slow cook-off) conditions and deflagrate under rapid heating (fast cook-off) conditions. Overestimating the heat flux may underestimate the hazard, motivating the need for physically-based methods that accurately predict heat flux from pool fires to embedded objects. In this chapter, the hazard is characterized in terms of the time to ignition of the explosive device and the violence (measured as kinetic energy of the exploded container) of the event.

Full-scale experimental investigation of heat transfer to objects in or near pool fires is limited because such experiments are expensive and difficult to instrument due to the harsh environment. Consequently, pool fire dynamics and heat transfer have been studied in small-scale, controlled laboratory settings, where detailed instrumentation yields high quality, quantitative data that is used to gain insight into the fire physics and the heat transfer process. Fire simulation tools based on computational fluid dynamics (CFD) offer a way to scale the laboratory experiments to larger, more realistic scenarios involving a variety of accidental conditions including wind speed and direction, size of the fire (1-100 m), and position of the object relative to the fire.

1.1 Chapter outline

Section 2 reviews the semi-empirical modeling approaches that have been emp loyed to estimate the radiation field from hydrocarbon pool fires. The fire community has used these approaches to provide immediate and practical engineering estimates of the radiation hazard. However, these approaches are unable to predict, a priori, the effects of changing fuels, wind conditions, and fire configurations.

Section 3 presents a framework for predicting heat transfer to embedded objects in pool fires based on a foundation of verification and validation (V&V). Sections 4-9 review next generation modeling tools for achieving high fidelity transportation fuel pool fire simulations within the V&V framework. Section 4 details how transportation fuels composed of complex mixtures can be represented by surrogate fuels that approximate the physical and chemical characteristics of the original fuel. Section 5 evaluates the capability of a chemical kinetic mechanism developed for such surrogate fuels to predict concentrations of soot precursors and outlines four method-ologies for calculating soot from its precursors. Section 6 discusses large eddy simulation (LES), a sophisticated numerical approach that captures the dynamics of buoyant pool fires. Section 7 describes a parameterization methodology (e.g. reaction model) for reducing the degrees of free-dom in detailed kinetic schemes of transpor tation fuel combustion. Section 8 discusses models that account for the complex and coupled interactions between turbulence and chemistry at the unresolved scale. Together, the reaction model and the model for turbulence/chemistry interac-tions allow complex combustion chemistry to be coupled to the LES simulation in a realistic way.



Heat Transfer to Objects in Pool Fires 71

Section 9 provides an overview of radiation, the dominant mode of heat transfer in most large pool fires, and its complexities as a spatial and spectral phenomenon.

Section 10 provides a brief overview of validation activities for heat transfer to embedded objects in transportation fuel pool fires. These activities focus on the use of a validation metric to quantify the level of agreement between experimental and simulation data. Section 11 illustrates the application of the LES fire simulation tool to the prediction of heat flux to an explosive device in a full scale hazards classification test for which data is unavailable. Section 12 demonstrates how an energetic material model can be coupled to the fire simulation tool to predict time to igni-tion of an explosive device. As the emphasis of this chapter is on predictive models, Section 13 concludes the chapter with a brief discussion on error quantification and predictivity.

2 Historical modeling approaches

2.1 Homogeneous flame

The early models of heat transfer from flames are based on the 1959 review by Hottel [1] of Blinkov and Khudiakov’s data on burning rate and flame height as seen in Fig. 1. The data include a number of fuels in pans with diameters ranging from 0.4 cm to 30 m.

The data were rationalized by equating the heat flux density, q≤, to the vaporization rate of the fuel, m· ≤, multiplied by the heat of vaporization, Δhvap. The heat flux to the fuel was decomposed into conduction, convection, and radiation contributions to give,

s -Δ = = - + - + - - .′′ ′′

4 4vap F o F o F o

4( ) ( ) ( )(1 e )KadK

m h q T T h T T F T Td

(1)

The first term on the right-hand side of eqn (1) represents conduction from the rim of the pan at the flame temperature, TF , to the liquid at To, where K is the liquid conductivity and d is the pan diameter. The second term represents convection from the flame to the liquid, where h is

Figure 1: Correlation by Hottel [1] of burning rate and flame height from pool fires as a function of pan diameter.




the convective heat transfer coefficient. The third term represents the radiation from the flame, where F is the view factor from the flame to the pan, s is the Stefan-Boltzmann constant, K is the absorption coefficient in the flame, and a is the ratio of the mean beam length to the pan diameter. This simplified model invokes the assumption of a homogeneous flame. Hence, a turbulent flame is assumed to have homogeneous gaseous and soot concentrations at some ‘effective radiation temperature’.

Hottel [1] established the framework for current semi-empirical models used to estimate radi-ation. For example, in Fig. 2 the fire is approximated by a cylinder at a uniform temperature and composition with a height HF , diameter DF , and temperature TF . Consider the flux per unit area, q·S, to an element at a distance RFS from the fire (eqn (2)),

4

FS F FS ,F Tq e s=

(2)

where

= , , ,FS F F FS( )F f H D R (3)

and

e e e e- = - - - .

2 2F CO H O soot(1 ) (1 )(1 )(1 )

(4)

To estimate the radiation from a homogeneous flame, one needs to know the flame shape and size to compute the geometric view factor, FFS; the flame absorption coefficient/flame emissivity, computed from both gas emissivities (eCO2

, eH2O) and soot emissivity (esoot); and an effective flame temperature, TF . Various semi-empirical approaches for estimating the radiation field in and around hydro carbon pool fires have been reviewed by De Ris [2] and Mudan [3]. A conical or cylindrical flame shape is usually assumed over a circular pool. A flame height can either be estimated through photographs or from the burning rate of the fuel. The nondimensional flame height (flame height to pool diameter ratio) has been found to correlate well with a nondimen-sional mass burning rate [3, 4]. Correlations relating the flame tilt angle from the vertical to wind velocity are also available [2].

Figure 2: Approximation of a fire by a homogeneous cylinder. Photograph taken by William Ciro, 2005.




Once the shape and size of the fire are calculated, the radiative characteristics of the fire need to be determined. The radiative properties of the flame are often estimated in the form of gray absorption coefficients or gray emissivities by assuming a homogenous mixture of CO2, H2O, and soot. Correlations are currently available for the spectral emissivities of combustion products of hydrocarbons [5]. The relative magnitude of CO2, H2O and soot emissivities are shown in Fig. 3 for partial pressures of CO2 and H2O of 0.12, a soot volume fraction of 10-7, a mean beam length of 3 m, and a flame temperature of 1,200 K. At 1,200 K, three quarters of the blackbody spectrum is in the 2.4-4.8 μm range, a range where soot radiation dominates. Therefore, the determination of soot emission and absorption is critical in computing accurate radiant heat fluxes from flames.

An alternative method of describing the fire hazard of a fuel is to estimate the total radiative out-put of the fire to its surroundings and report that radiative output as a fraction (cR) of the total heat of combustion. This fraction cannot be determined theoretically and is normally estimated [2].

2.2 Homogeneous model and observable fire phenomena

The major shortcoming of the homogeneous model is the evaluation of the effective flame tem-perature. In his review, Hottel assumed a value of 1,100 K [1]. The effective flame temperature, however, is dependent on pool size and to a lesser degree on fuel type [6, 7]. Figure 4 shows average surface emissive power as a function of pool diameter for a range of fuels. In general, the radiation is found to increase with pool diameter as a result of the increase in emissivity. However, for large pool fires, the radiation decreases as a result of the shielding of flame radiation by the outer, cooler soot layers. This phenomenon is evident in Fig. 4, where several fuels show an effective emissive

Figure 3: Spectral emissivities of CO2 (pCO2 = 0.12), H2O (pH2O = 0.12), and soot (volume

fraction = 10-7) at 1,200 K for a mean beam length of 3 m.




power of the flame surface passing through a maximum at pool diameters of 1-10 m, with peak val-ues near 150 kW/m2. Liquid natural gas (LNG) is the exception; being lightly sooting, its shielding effects are not yet evident. For this reason, the maximum radiation for large LNG flames exceeds that of more sooting fuels. In the review by Mudan and Croce [8], peak emissive power values of 220 kW/m2 are reported for land-based LNG fires (higher values are found on water) compared with peak values of 160 kW/m2 for LPG and 130 kW/m2 for gasoline.

The shielding of the core of the flame by soot has been studied for some time, with Smith [9] first proposing models that tried to provide a mathematical framework for the observations of the periodic transport to the surface of large eddies from the hot core. The emissive powers of larger flames vary widely due to this phenomenon. Mudan [3] estimated that the luminous zones covered 20% of the surface of the flame and had emissive powers in the range of 110-130 kW/m2, while the cooler background had an emissive power of 20 kW/m2. Thermographic cameras have been used by Schönbucher’s research group to obtain time-resolved measurements of the

Figure 4: Average surface emissive power of pool fires for different fuels as a function of pool diameter [6].




emissive power distribution in flames. They have developed probabilistic models that describe hot spots with surface emissive powers ranging from 33 to 430 kW/m2 and colder soot parcels with surface emissive powers ranging from 6 to 50 kW/m2 [7].

Radiative fluxes to the pool surface and even at locations away from the fire are likely to be influenced by the assumed flame size and shape, quantified by geometric view factors [10]. Numerical estimates of the radiative heat fluxes to the pool surface from 30 cm diameter pool fires employing the homogeneous model were found to be higher than the experimental values by 40% [2]. Most of this error was attributed to assuming a conical shape to the flame.

Another shortcoming of the homogeneous model is its inability to predict the radiative feedback to the pool surface, particularly in large pool fires. Obtaining accurate estimates of the radiative fluxes to the pool surface is important for determining fuel burning rates. Hottel [1] was able to explain the trends in the burning rates of liquid fuels by relating the rate of heat transfer from the fire to the pool to the rate of fuel vaporization, but the effective emissive power of the flame that he assumed was, in effect, a fitting parameter. The cooler, unburned, sooty pyrolysis gases near the fuel surface in large fires may block part of the flame radiation from reaching the surface, similar to the effects observed for external heat transfer. Shinotake et al. [10] showed that radiation block-age significantly affects the fuel burning rates in pool fires of diameters greater than 1 m. They also observed that the experimentally measured radiative fluxes to the pool increased with increase in diameter but then quickly saturated compared to the external fluxes. They explained these observa-tions in terms of radiation blockage by performing simple two-layer model calculations assuming conical shapes. An outer cone represented the radiative characteristics of the fire and an inner cone represented the vapor dome of pyrolysis gases. The assumption of a homogeneous flame failed to capture the observed trends in heat fluxes. However, the two-layer model calculations were found to be very sensitive to the adopted soot concentrations and soot temperatures in the flame as well as to the vapor dome. Measurements in very large pool fires also show significant gradients in the radiative heat fluxes to the pool surface, which are likely to result in significant gradients in the fuel vaporization rates within the pool [11].

In Fig. 4, the mean surface emissive power for many hydrocarbon pool fires is seen to decrease with increasing pool diameter due to smoke obscuration. Although a systematic methodology to reliably address this phenomenon is not yet available, some explanations have been proposed. The vapor dome of large fires may contain pyrolyzed fuel vapors which are at moderate tem-peratures relative to the reaction zone. Poor mixing and/or the slow entrainment of this stream with the air stream may result in the formation of long-lived, fuel-rich eddies that contain unox-idized fuel [12]. The smaller fluid strain rates associated with this process can reduce the diffu-sion rates, giving the fuel more time to pyrolyze and to form larger soot particles (smoke) that take longer to oxidize.

Klassen and Gore [4] measured transient emission and absorption properties in pool fires of different fuels and sizes (maximum diameter of 1 m). They observed a relatively cold layer of soot particles near the fuel surface. Comparing their absorption and emission measurements, they showed that a large portion of the soot particles were at relatively low temperatures and did not contribute to emission. Therefore, it is important to understand both the chemical phenomena which lead to the formation of soot, and the local transport phenomena which determine the distributions of soot and soot temperature within a flame. The local soot concentration results from a time evolved history of local production and oxidation as well as convective and diffusive (thermophoretic) transport processes [13]. In fact, in laminar diffusion flames, the peak soot concentrations have been found to be slightly offset from the location of peak temperature [14]. This phenomenon is shown in Fig. 5 for a laminar C2H2 diffusion flame above a burner with a 12 mm × 96 mm fuel slot.




The local radiant emission from a flame is linearly dependent on the soot concentration and is dependent on temperature to the fourth power. The effective emissive power at the flame surface is the integral of the local emissive power multiplied by the transmissivity to the surface and cor-responds to an emission temperature that is intermediate to the maximum flame temperature (~1,960 K) and the temperature at the position of maximum soot concentration (~1,640 K). Hence, knowledge of the temperature and soot volume fraction distributions is critical in calculating the effective flame temperature across flame fronts. In pool fires, similar effects occur on a macro-scopic level due to the shielding of the flame core by the cooler, external soot layers and at a microscopic level as a consequence of the soot radiation from flamelets in the combustion zone.

The maximum heat flux is normally used to calculate safe separation distances from fires using metrics on damage from radiation such as those provided in Fig. 6: heat flux that causes pain to exposed humans, yields skin burns, or ignites wood for different times of exposure. Maximum tolerable heat fluxes can be established for different assumed times of exposure. Soot obscuration of radiation from fires will result in an overestimation of heat flux if flame tempera-ture is assumed to be independent of diameter. This error will result in a conservatively safe distance of separation.

For certain problems, however, overestimation of the heat flux may underestimate the fire hazard. This is particularly true when containers of high energy materials are exposed to radia-tion. An example of how lower heat fluxes can lead to greater hazards is shown in Fig. 7, where time to explosion of containers of the explosive PBX is plotted as a function of heat flux to the container surface. As expected, the time to explosion increases as heat flux decreases. However, at low heat flux rates, the intensity of the explosion increases as shown by the inset figures of the remnants of the container for two heat flux levels. The violence of the explosion increases as the time to explosion increases. Indeed, it is well known in the explosives community that long heat-ing times (e.g. slow cook-off) can lead to detonations since more of the explosive material is heated to the ignition temperature. In contrast, with fast heating times (e.g. fast cook-off) only a surface layer is heated to the ignition temperature. The problem of BLEVEs with LPG is influ-enced by the accumulation of energy in the storage tanks that leads to the greatly enhanced strength of the explosions that result. For these reasons, conservative estimates of heat flux are no longer adequate.

Figure 5: Radial profiles of soot concentration and temperature at an axial height of 7.14 mm in a laminar C2H2 diffusion flame (Fig. 27 of ref. [14]).




Fire modeling approaches more sophisticated than the homogeneous model are required to reliably address observed pool fire phenomena. These phenomena, including the effects of fuel type, smoke obscuration, relative locations of the flame front and of regions of high soot concen-tration, variation of local flame temperature, radiation blockage, and radiative feedback to the

Figure 6: Skin exposure times to different heat fluxes that result in pain or burns, and flux needed to ignite wood (adapted from [8]).

Figure 7: Time to ignition for containers of the explosive PBX as a function of heat flux, using both electrical heating and external flames. Inset figures show the recovered container fragments at two heating rates.




pool surface determine the radiative heat transfer to an embedded object [15, 16]. The past two decades have seen an increasing use of CFD-based models to study fire phenomena. The prog-ress that has been made will be discussed in the remainder of this chapter.

3 V&V as a foundation for predicting heat transfer to embedded objects in pool fires

The goal of this chapter is to present a physically-based method for predicting the potential hazard of an explosive device immersed in or near a pool fire of transportation fuel. To accomplish this goal, CFD-based computational tools that capture the relevant physical processes associated with the fire and the heat-up of the explosive device are employed.

To move toward predictivity with these computational tools, we choose a methodology based on the V&V principles set forth by Oberkampf and Trucano [17]. Verification is the process of determining whether or not the mathematical models are implemented into computer code as the programmer intended, independent of the model’s physics. Validation determines how well the computer model matches the physical world. The process of V&V is cyclical as shown in Fig. 8, involving development of the conceptual model, verification of the model implementation, vali-dation of the physical results, and evaluation of the conceptual model. Certification of the com-puter code for predictive use and quantification of error in the prediction involves the two-way coupling between the various stages of the V&V cycle.

3.1 V&V hierarchy

A key tool in the V&V methodology is the construction of a V&V hierarchy. The apex of the hierarchy is the specific intended use of the simulation tool, i.e. the full system to be simulated. The remainder of the hierarchy is composed of several levels of decreasing technical complex-ity: subsystem cases, benchmark cases, coupled problems, unit level problems, and molecular processes. As one moves down the hierarchy, the quantity and quality of data increases and the experimental uncertainties decrease.

Figure 8: Connectivity between verification, validation, simulation, and certification (adapted from [18]).




At the highest level of the hierarchy, data are directly applicable to the intended use of the model but limited in scope and accuracy and often qualitative in nature. The subsystem case level is the decomposition of the overall system into simpler systems. Data with high experimental uncertainties are generally available for subsystem cases. The third level consists of benchmark cases, in which detailed experimental data from simplified but fully coupled problems are avail-able for comparison. Coupled problems at the fourth level consist of two or more unit problems coupled together. Data available at this level include standard numerical solutions to simple problems and experimental data for coupled systems. The unit problems at the fifth level consist of isolated physical models. These models are validated as stand-alone problems with highly accurate numerical and experimental data. The lowest level in the hierarchy, molecular pro-cesses, further divides the unit problem into its fundamental components. The high fidelity data available at the unit problem level is also available at this level.

A V&V hierarchy is constructed for heat transfer to explosive objects embedded in transpor-tation fuel pool fires with V&V activities from the molecular processes level to the full system level as seen in Fig. 9. Some of these activities are highlighted in subsequent sections to dem-onstrate this hierarchal approach in moving toward predictivity. With one exception, the boxes at each level represent cases where experimental data sets have been identified in the context of the intended use of the simulation tool. The one exception is the ‘buoyancy driven flames’ box at the benchmark level; the desired experimental data for this box is unavailable.

3.2 Validation metric

Validation of computed results in simulation science has generally consisted of comparing in graphical manner form (e.g. a two-dimensional plot) data extracted from a simulation or set of simulations with measured observables from an experiment or set of experiments. Based on such

Figure 9: V&V hierarchy for simulations of heat flux to an object embedded in a transportation fuel pool fire.




a graphical comparison, a person may declare that the computer model is ‘validated’, ‘invalidated’, or ‘requires improvement’. However, as pointed out by Oberkampf and Barone [19], these state-ments are qualitative in nature, leaving conclusions to the discretion of the observer. This poten-tial for widely varying conclusions creates the need for a non-biased measurement, or metric, for determining the ‘level of agreement’ between experimental evidence and simulation results. The objective numerical values provided by the metric can then be used to formulate a value judgment of the comparison based on the level of risk one is willing to accept for the intended application. While the value judgment still requires the intervention of the biased human, the cal-culation of the metric does not include a notion of ‘quality’, thus making the metric itself a non-biased participant in the validation process. Here, we briefly review a metric based on the use of statistical confidence intervals. This metric will be used in subsequent sections of this chapter to evaluate the level of agreement between LES data and experimentally measured data.

Given a set of two or more experimental observations (n ≥ 2) and assuming that a population can be described by a normal distribution, a confidence interval for the true mean, μ, can be con-structed using degrees of freedom ( = n -1) as

(5)

where x is the observed mean, s is the standard deviation, the level of confidence is 100(1 - a)%, and ta/2, is the student-t distribution value based on the values of a and .

Now, given a set of simulation and experimental data, one may construct an estimated error,

m e ,E y y= -

(6)

where ym refers to the model (simulation) results and __ y e refers to the sample mean of the experi-

mental results, e.g. the average of a set of n experimental observations. Note that E is referred to as the ‘estimated’ error rather than the actual error because the true mean (μ) cannot be known given a small set of observations. Next, the true error is written as

m – .E y m= (7)

Using the above definitions, the expression for the confidence interval for the true error is,

(8)

When computing metrics over a range of input variables such as spatial location, it is useful to reduce the collection of metrics to a single, global metric representing the system. The average relative error metric and the average relative confidence indicator, as described by Oberkampf and Barone [19], can be used to compute a global metric. The average relative error metric is defined by

(9)

where xu and xl are the upper and lower bounds of the input variable. The relative average confi-dence indicator is given by

(10)




The average relative error metric and the average relative confidence indicator provide a compact

statement regarding the level of agreement, expressed as the global metric ±

e e aveave/ / .E y CI y

A few important points need to be considered relative to the metrics outlined here. First, the confidence intervals for the mean are not to be interpreted as traditional error bars on the simula-tion results; rather, they are the intervals in which the true error is estimated to lie with a given confidence. Second, for small numbers of experimental observations, the confidence intervals are made larger by increa sing values of ta/2, . It is thus advantageous to have multiple replications of a given experiment. Third, the confidence intervals speak to the quality of the experi mental data and may lead one to seek out more and/or improved data sets. Finally, experimental observations are given supremacy over simulation results, as acknowledged by the sole use of experimental data in the construction of the confidence intervals.

4 Surrogate fuel formulation

The use of CFD to model heat flux to an explosive device in a transportation fuel pool fire raises the challenge of how to represent complex hydrocarbon fuels in computer simulations. One example of a transportation fuel is jet fuel (Jet-A or the military counterpart, JP-8), a mixture of hundreds of hydrocarbons that varies geographically and with time. It would be impractical to perform a simulation with such a complex mixture, even if the thermodynamics and detailed chemical kinetics for all the species in the mixture were available. Therefore, surrogate fuels, for which the necessary chemical and physical characteristics are known, must be developed. Differ-ent surrogates may be formulated for a given fuel depending upon the flame properties of inter-est. In this section, surrogate fuels are formulated for use in the simulation of a jet fuel pool fire, with particular interest in matching the burning rate and the heat transfer to objects immersed in, or in close proximity to, the fire. This work corresponds to the ‘surrogate fuel formulation’ box at the molecular processes level of the V&V hierarchy in Fig. 9.

The major categories of hydrocarbons in jet fuels are normal and branched alkanes, cyclo-alkanes, and aromatics. Several investigators have developed surrogates for jet fuels [20-24] for applications other than pool fires. For the application of a transportation fuel pool fire, each surrogate component is required to have known chemical kinetics, to be representative of a main class of hydrocarbons present in jet fuels, and to be relatively inexpensive. In order to match burning rate and heat transfer to embedded objects in a pool fire, the mixture of components must match the volatility of the fuel, the sooting tendency, and the heat of combustion, and must reproduce the flame characteristics of a Jet-A/JP-8 pool fire, preferably with a small number (<10) of components.

For the surrogate fuel study, the fuels tested included Jet-A, Norpar-15, and surrogates com-posed of various chemical reagents. Jet-A was acquired from the Salt Lake City Airport. Nor-par-15, obtained from Exxon Chemicals, is a narrow-boiling range mixture of hydrocarbons. Three surrogates, Hex-11, Hex-12, and Hex-25c, each composed of six compounds, were formu-lated to match volatility (the boiling point distribution) and sooting tendency (smoke point) of the Jet-A/JP-8 fuel. The compositions of all five fuels are listed in Table 1.

Other properties of the Jet-A and the calculated properties of the three surrogates are listed in Table 2. All three surrogates are similar to Jet-A in terms of volatility as reflected by the flash point and the average boiling point and in terms of smoke point.

4.1 Validation of surrogate formulation

The ability of the surrogate formulations to match the burning rates and heat fluxes in Jet-A/JP-8 pool fires was tested by burning the jet fuel and its surrogates in a round steel pan, 0.3 m in diameter




and 0.1 m deep, placed 0.5 m above the ground. The tests were conducted in an enclosure 5 m × 5 m in cross-section and 6 m high equipped with dampers to control air infiltration and an exhaust duct. Both transient and steady-state tests were performed. Transient tests consisted of igniting and burning a batch of fuel in the pan. Steady-state tests involved continual replacement of the fuel in the pan to maintain a constant fuel level. Flame heights, shape, and puffing frequency were deter-mined with a high-speed video camera shooting at 2,000 frames/s and with real-time video. Total heat fluxes and radiative heat fluxes were measured with gas-purged, water-cooled radiometers. Details of the test conditions can be found in [25, 26].

In classical studies of hydrocarbon pool fires [27-29], the mode of pool fire combustion (e.g. transient or steady-state) is not always stated. The tests described here indicate that the combus-tion mode plays a significant role in the measured fuel properties of interest.

Table 1: Fuel compositions (in mol. % except where noted).

Hex-11 Hex-12 Hex-25c Norpar-15 Jet-Aa

n-C8, 3.5 n-C8, 3.0 n-C8. 5.0 n-C14, 34.4 n-Paraffin, ≈28n-C12, 40.0 n-C12, 30.0 n-C12. 32.0 n-C15, 49.0n-C16, 5.0 n-C16, 12.0 n-C16, 13.5 HMNb, 10.0 n-C17+, 3.1 Branched paraffin, ≈29Xylenes, 8.5 Xylenes, 15.0 n-HxBec, 23.0 Mono-aromatics, ≈18Tetralin, 8.0 Tetralin, 13.0 PMHd, 15.0 Di-aromatics, ≈2Decalin, 35.0 Decalin, 27.0 Decalin, 15.0 Cycloparaffin, ≈20 Nondetermined, ≈3Sum, 100.0 100.0 100.0 100.0 100.0a Approximate composition of Jet-A in this table is in wt. %.b 2,2,4,4,6,8,8-heptamethylnonane.c n-hexylbenzene.d 2,2,4,6,6-pentamethylheptane.

Table 2: Properties of Jet-A and surrogate fuels.

Properties Jet-A Hex-11 Hex-12 Hex-25c

Smoke point (mm) 24.5 28.7 23.1 24.0TSIa 26.7 17.6 22.1 20.3MW (g/mol) 173.5b 151.5 152.2 166.5VABPc (°C) 220.2 211.1 215.7 209.5Flash point (°C) 40.9 40.3 41.3 39.0Latent heatd (kJ/kg) 254.6 280.4 281.8 253.8Combustion heat (MJ/kg) 44.9 44.5 44.6 44.6aTSI is the threshold soot index. It is defined based on the smoke point such that its value ranges from 0 (least sooting) to 100 (most sooting).bMW (molecular weight) of Jet-A is estimated using the API empirical equation.cVABP is the volumetric average boiling point. It is the mean of the 10%, 30%, 50%, 70%, and 90% recovery tempera-tures determined in ASTM D86.dLatent heat is estimated at VABP.




4.2 Burning rates and heat fluxes at steady state

In the steady-state experiments, the liquid fuel density was assumed to be constant throughout the test. Gas chromatograph (GC) spectra of fuel samples taken from the burning pan show that Jet-A samples did not change in composition over time.

Instantaneous volumetric burning rates were calculated and averaged over a time interval of 60 s. The results for Jet-A are plotted in Fig. 10 as a function of time. For this experiment, the steady-state burning rate of 2.07 × 10-3 m/min (0.0278 kg/m2 s) is reached 24 min after ignition.

For liquid pools greater than 0.2 m in diameter, the mass burning rate (m≤, kg/m2 s) can be predicted by

[1 exp( )]m m k db∞= - - ⋅ ⋅′′ ′′ (11)

where m≤∞ is the mass burning rate of an infinite diameter pool (kg/m2 s), k is the extinction-absorption coefficient of the flame (1/m), b is the mean beam length corrector, and d is the pool diameter (m) [30]. As there are no reported constants for Jet-A pool fires, values of m≤∞ and k · b for kerosene are used (0.039 kg/m2 s and 3.5 m-1, respectively) [28]. From eqn (11), the com-puted mass burning rate for Jet-A is 0.0258 kg/m2 s (1.91 × 10-3 m/min), which is close to the experimental value for Jet-A reported above and to the value of 1.9 × 10-3 m/min reported for a 30 cm kerosene pool fire [1, 27].

Results from Hex-12 and Norpar-15 tests are also plotted in Fig. 10 for comparison. The steady-state regression rates for Hex-12 and Norpar-15 are 1.90 and 0.96 × 10-3 m/min, respectively. The time required for Hex-12 to reach steady state in this configuration is approximately 24 min.

For the intended use of heat transfer to an explosive device, a key measure for the surrogate is its ability to yield radiation intensities that match those from a Jet-A flame. Real time heat flux mea-surements are shown in Fig. 11 for Jet-A and Hex-12 flames. These measurements were taken at a

Figure 10: Continuous feed, constant level 30 cm diameter pool fire surface regression rate profile.




height of 0.20 m above the fuel surface and 0.40 m from the center of the pan. About 20 min after ignition, the radiative heat flux reached a relatively constant value of 10.9 kW/m2 for Jet-A and 11.4 kW/m2 for Hex-12. The average radiative heat flux for Norpar-15 was 5.8 kW/m2.

4.3 Burning rates and heat fluxes for transient burning

In a typical pool fire, a fixed quantity of fuel is ignited and burned to completion. Due to the com-plex mixture of hydrocarbons present in Jet-A/JP-8, transient behavior is observed for many of the key physical and chemical parameters. Lighter hydrocarbons are expected to vaporize and burn preferentially such that during the later stages of the fire, the fuel should be enriched in residual heavy hydrocarbons. The transient experiment was conducted by filling the pan with fresh fuel, then turning off the feed system and allowing the fuel to burn to completion. The decreasing fuel level as a function of time, used to compute the surface regression rate, was measured using an optical level sensor [25].

The surface regression rate of Jet-A in transient tests is shown in Fig. 12. The surface regres-sion rate increases rapidly up to 10 min and reaches a peak value of 1.84 × 10-3 m/min at 11 min. This peak value is close to the burning rate obtained in the steady-state tests. The rate falls off rapidly over the range of 12-35 min and then decreases more slowly from 40-80 min until the end of burning. The Jet-A burning rate profile is successfully simulated by the surrogate fuel Hex-12 as shown in Fig. 12. The peak burning rate of Hex-12, 1.64 × 10-3 m/min, is slightly lower than that of Jet-A. The average burning rate for Hex-12 is 0.77 × 10-3 m/min, slightly lower than the average of 0.82 × 10-3 m/min for Jet-A.

Two explanations have been proposed for the high burning rate soon after ignition [31]. First, in the initial burning stage, there is no heat loss to the edges of the fuel pan and the entire amount of heat transferred back from the flame to the pool surface can contribute to fuel vaporization.

Figure 11: Radiative heat flux measurements from 30 cm diameter pool, continuous feed, constant fuel level experiments.




Second, light components burn much faster initially, and the burning rate decreases after these species are depleted [28, 29].

The issue of whether the high initial burning rate is due to intense thermal feedback or to com-positional variation was addressed by conducting transient burning experiments with Norpar-15, which is composed primarily of normal alkanes with 14-16 carbons (>99.8%) as shown in Table 1. After an initial transient, the surface regression rate is relatively constant, which supports the view that the initial high boiling rate for Jet-A (and its surrogates) is due to compositional change. Further confirmation was provided by the agreement of the mean fuel regression rate for Nor-par-15 in the transient and steady state (Fig. 10) experiments. The radiant heat flux from the flame to the surroundings was consistent with the change in burning rate as well [25]. Finally, direct evidence of compositional change over time was demonstrated by GC analysis of Jet-A samples taken from the fuel pan [25].

4.4 Effect on fuel composition changes on sooting propensity

The composition changes described above are expected to lead to a continuous change in smoke point. Based on an ASTM D86 distillation test [32] of the Hex-12 surrogate, smoke points were calculated at different stages of volume loss and the results are plotted in Fig. 13. In addition, smoke points were measured on samples taken in a transient pool fire at different percentages of fuel volume consumption. Smoke points for Hex-12 increase with increasing volume loss for samples from both pool fire tests and distillation tests because the soot-promoting compo-nents (xylenes, tetralin, and decalin) are more volatile than n-dodecane and n-cetane. By the end of a burn, the residual surrogate fuel is nearly pure n-cetane and reaches its highest smoke point. In contrast, the smoke point of Jet-A decreases slowly with increasing burn-off/boil-off. This decrease is believed to be related to the wide spectrum of aromatics in actual jet fuel, as suggested in the detailed hydrocarbon analysis for Jet-A [33]. The existence of high molecular

Figure 12: Transient surface regression rate profile for 30 cm diameter pool fire.




weight, high boiling point, and high sooting index naphthalenes and benzo-cycloalkanes in the actual fuel helps to maintain the smoke point relatively constant through low and intermediate percentages of fuel consumption with a slight decreasing trend at high percentages of fuel con-sumption.

4.5 Improved surrogate formulation

Due to the challenges associated with matching sooting propensity of Jet-A over its entire boil-ing range (or lifetime of a transient pool fire), a more chemically complex surrogate is required. The method of structural group contributions was adapted to the formulation of surrogates [25] and was used to provide a more chemically accurate description of the fuel [26]. The method is a significant improvement over previous approaches as it does not require any experimental proce-dures or information on fuel properties. However, the molecular structure of the fuel molecules must be determined.

The improved surrogate, Hex-25c, consists of six species. Its composition is given in Table 1, while its physical properties are given in Table 2. A comparison of the smoke point of Hex-25c with that of Jet-A is shown in Fig. 13, where it is evident that the new surrogate provides significant improvement in smoke point performance over the Hex-12 surrogate, particularly in the later stages of fuel consumption.

5 Chemical kinetics for soot production from JP-8

Two other areas of major research activity at the molecular processes level are the kinetic model-ing of the gas and the solid (soot) phases. Kinetic modeling of surrogate fuels requires mecha-nisms and chemical kinetics for the major components of the surrogate [20-24, 34, 35]. Although the kinetic mechanisms are still under development, considerable success has been achieved by various research groups in matching experimental results. In general, these reaction mechanisms

Figure 13: Smoke point variation as a function of volume of fuel burned (or distilled) for jet fuel and surrogates.




are tuned to meet specific objectives. For example, one focus area in engine modeling is ignition, which is influenced by chemical kinetics at low temperatures.

For the purpose of calculating heat transfer to embedded objects in pool fires, soot forma-tion, which is strongly dependent on soot precursors derived from acetylene and benzene, must be predicted. The following section describes kinetics that have been optimized to predict ben-zene and acetylene in premixed flames. The resulting mechanism is called the Utah Surrogate mechanism [36].

5.1 Utah Surrogate mechanism

A schematic representation for soot formation, adapted from Bockhorn [37], is shown in Fig. 14. The initial steps are the ignition and consumption of the surrogate mixture, which is driven by reac-tions with H, O, and OH radicals (with contributions from HO2 in the ignition zone). The larger paraffinic fuel molecules decompose and cascade down to the smaller aliphatic and olefinic molecules. The key pathway to soot is the formation of benzene and then polycyclic aromatic hydrocarbons (PAH), which are the building blocks of the first particles. Acetylene is important because it is the major contributor to soot mass addition and it participates in the H-abstrac-tion-ACetylene-Addition (HACA) mechanism [38] that leads to the growth of PAH and soot. Reaction mechanisms that address the steps shown in Fig. 14 are large, involving hundreds of chemical species and thousands of chemical reactions [22-24, 36-41].

The Utah Surrogate mechanism is formulated from detailed sub-models of n-butane, n-hexane, n-heptane, n-decane, n-dodecane, n-tetradecane, and n-hexadecane; semi-detailed sub-models of i-butane, i-pentane, n-pentane, 2,4-dimethyl pentane, i-octane, 2,2,3,3-tetramethyl butane, cyclo-hexane, methyl cyclohexane, tetralin, 2-methyl 1-butene, and 3-methyl 2-pentene; and aromatics

Figure 14: Schematic representation of sequential steps for soot formation and burn out (adapted from [37]).




that include benzene, toluene, and xylenes. The mechanism is available as supplemental material provided to the Combustion Institute in support of the publication by Zhang et al. [36]. The mech-anism can be used to predict the fuel consumption and major combustion products for jet fuels that are comprised of mixtures of n-paraffins, i-paraffins, cyclo-paraffins, aromatics, and alkyl-substi-tuted aromatics. Jet fuel composition can vary widely, depending upon the crude oil being refined and the refining procedures used. For example, aromatic content varies from 11% to 26%.

The mechanism was built on the following foundation:

Marinov• -Westbrook-Pitz hydrogen model [42]Hwang • et al. [43] and Miller et al. [44] acetylene oxidation modelsWang and Frenklach acetylene reaction set with vinylic and aromatic radicals [45]•Marinov and Malte ethylene oxidation sub-model [46]•Tsang propane and propene chemical kinetics [47, 48]•Pitz and Westbrook • n-butane sub-model [49]Miller and Melius benzene formation sub-model [50]•Emdee• -Brezinsky-Glassman toluene and benzene oxidation sub-model [51]Vovelle and coworkers • n-heptane decomposition model [52]Pitsch • i-octane decomposition model [53]

New submodels were added for a number of n-paraffins (C5, C6, C10, C12, C16), a number of iso-paraffins (i-C4, i-C5, i-C6, 2,2,3,3,-tetramethyl-C4), cyclohexane, methyl cyclohexane, buta-diene, and 3-methyl 2-pentene.

The mechanism is able to model a wide range of surrogates. It has been optimized for atmo-spheric conditions, flame studies, and soot precursors. Its ability to model the concentration of acetylene in flames of common components of surrogates, as well as a kerosene flame, is shown in Fig. 15, where results for acetylene concentration as a function of height above the burner are presented. The conditions for the flames are given in Table 3. Kinetic model predictions for the pure component fuels and for kerosene using a surrogate formulation are shown by the solid lines. The agreement with the data shows the progress that is being made in the development of kinetic models with predictive capabilities. Nevertheless, a validation metric has not been applied to this analysis, so quantitative information is not available. Similar comparisons for benzene concentration can be found in [36].

5.2 Soot formation and oxidation

The Utah Surrogate mechanism provides the gas phase reactions up to the point of particle inception, as shown in Fig. 14. Several soot models are available that can provide the transition from the soot precursor molecules to soot. Four classes of models utilized in order of increasing complexity are:

Empirical models like that of Sarofim and Hottel [63] represent a combination of soot forma-•tion and destruction. Such models account for some of the major factors affecting soot contri-butions to radiative absorption and emission: temperature, stoichiometric ratio, and fuel type. With empirical models, a certain fraction of the fuel is converted to soot under fuel rich condi-tions, but no soot persists in a fuel lean environment.The Lindstedt model [39] has four steps: nucleation, surface growth, oxidation, and coagula-•tion. The nucleation is assumed to be an activated process that is proportional to acetylene




BockhornC2H2 Flame

0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

0 1 2

(a) (b) (c) (d)

(e) (f) (g) (h)

3HAB (cm)

Mol

e F

ract

ion

BastinC2H2 Flame

0.0E+00

1.0E-01

2.0E-01

3.0E-01

0 1 2 3HAB (cm)

Mol

e F

ract

ion

MarinovCH4 Flame

0.0E+00

1.0E-02

2.0E-02

3.0E-02

0 0.2 0.4 0.6 0.8HAB (cm)

Mol

e F

ract

ion

CastaldiC2H4 Flame

0.0E+00

2.0E-02

4.0E-02

6.0E-02

0 0.4 0.8 1.2HAB (cm)

Mol

e F

ract

ion

BittnerC6H6 Flame

0.0E+00

2.0E-02

4.0E-02

6.0E-02

0 0.5 1 1.5 2HAB (cm)

Mol

e F

ract

ion

VovellenC71.0 Flame

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

0 0.05 0.1 0.15HAB (cm)

Mol

e F

ract

ion

VovellenC7 Flame

5.0E-03

1.0E-02

1.5E-02

2.0E-02

0 0.1 0.2 0.3 0.4 0.5HAB (cm)

Mol

e F

ract

ion

VovelleKerosene

Flame

0.0E+00

5.0E-03

1.0E-02

1.5E-02

2.0E-02

0 0.1 0.2 0.3HAB (cm)

Mol

e F

ract

ion

0.0E+00

Figure 15: Comparison of concentrations of acetylene simulated using the Utah Surrogate mechanism with experimental results reported by different authors (see Table 3 for experimental conditions): (a) atmospheric pressure fuel rich flames of acetylene (C2H2), (b) low pressure fuel rich flames of C2H2, (c) fuel rich flames of methane (CH4), (d) fuel rich flames of ethylene (C2H4), (e) fuel rich flames of benzene (C6H6), (f) stoichiometric flames of n-heptane (nC71.0), (g) fuel rich flames of n-heptane (nC7), and (h) fuel rich flames of kerosene.

Table 3: Experimental conditions for premixed flames in Fig. 15.

Inert Flow rate Author Fuel Ar (%) C/O P (torr) (g/(cm2 s)) Reference

Marinov CH4 0.453 0.626 760 7.19 × 10-2 [54]Westmoreland C2H2 0.05 0.959 20 1.58 × 10-2 [55]Bastin C2H2 0.45 1.00 19.5 3.46 × 10-2 [56]Bockhorn C2H2 0.55 1.103 90 3.43 × 10-2 [57]Harris C2H4 0.656 0.92 760 1.12 × 10-1 [58]Castaldi C2H4 0.578 1.02 760 7.21 × 10-2 [59]Bittner C6H6 0.3 0.717 20 2.19 × 10-2 [60]Ciajolo C6H6 0.752a 0.72 760 5.07 × 10-2 [61]Vovelle C7H16 0.73a 0.605 760 6.50 × 10-2 [62]Vovelle i-C8H18 0.682a 0.608 760 5.56 × 10-2 [62]Vovelle C10H22 0.682a 0.558 760 6.68 × 10-2 [34]Vovelle Kerosene 0.684a f = 1.7 760 7.96 × 10-2 [34]aN2 is used the inert rather than Ar.




concentration and which yields nuclei of a specified size, with 100 carbon atoms being sug-gested. Surface growth is proportional to acetylene concentration and has a rate parameter fitted to literature rates. Soot is assumed to be oxidized by O2 with a rate constant fitted to data so as to allow for the role of OH. Coagulation is calculated using the standard equations for aerosol dynamics.The HACA mechanism, built on the two-step process of activation of an aromatic molecule •by hydrogen abstraction followed by acetylene addition, leads to both molecular weight growth and cyclization [40]. Lumping mechanisms have been developed for the PAH growth using generic rates for the different classes of reactions including acetylene addition, hydro-gen abstraction, and reactions with OH and O2. Soot nucleation is assumed to occur by the dimerization of two PAHs. Although dimers of all PAH combinations can be included, it is common to use pyrene. Soot formation and growth are calculated using moment methods, with allowance for nucleation, coagulation, and surface growth, as well as oxidation reac-tions. The implementation of this method by Appel et al. [38] has found widespread use.Sectional models treat the soot simultaneously with the chemical kinetics by assigning •large PAH particles and soot particles to bins that have a range of carbon numbers. The transition from gas phase chemistry to particle chemistry is achieved by assigning a bin with a given carbon number as the smallest particle size. One implementation of the sec-tional model [41] has the smallest bin for mass numbers of 201-400, with an H/C ratio of 0.5, corresponding to a particle size of 0.85 nm. The mass limits are approximately doubled for each sequential bin; the largest of 20 bins corresponds to mass numbers of 105-210 million, an H/C ratio of 0.125, and a particle size of 68 nm. Bins are assumed to react with all other bins and with gas phase molecules. The first four bins are considered to be large PAHs and the bins from 5 to 20 to be soot particles. Soot can also be oxidized, primarily by reactions with OH, O, and O2. The rates of oxidation for most flame conditions are domi-nated by the reaction with OH. The rate of reaction is proportional to the collision rate of the OH with the soot surface with approximately 13% of the collisions [64] leading to the consumption of soot by the stoichiometric reaction C + OH = CO + H.

6 Use of LES methods for pool fires

A major challenge in applying CFD to fires is the wide range of continuum length scales and their corresponding time scales that characterize the fire physics in large diameter (>1 m) fires. For example, important physical time and length scales range from molecular O(10-9 s, 10-10

m) to scales that are observable with the naked eye O(1 s, 1 m). This range of time and length scales prohibits the use of fully resolved, three-dimensional, direct numerical simulation (DNS) techniques. Additionally, transportation fuel fires often involve complicated interactions with the environment such as the highly unsteady processes of fluid/structure interaction, wind effects, and flame spread across fuels.

Given current modeling options and the importance of unsteady effects in transportation fires, LES is the prime candidate for modeling such fires. Compared to the traditional Reynolds aver-aging (RANS) approach, LES captures the unsteady effects of pool fires more accurately by resolving the large length and time scales that are responsible for controlling the dynamics of the fire [65]. In fact, LES is emerging as the prevailing methodology for studying fires due to its abil-ity to render realistic, time-resolved flows of gases, heat, and smoke throughout a domain [66].

An LES approach was employed by Schmidt et al. [67] and Kang et al. [68] to study turbulence structure in medium scale methanol pool fires. In both these efforts, reasonably good agreement




was obtained for the mean velocity and temperature fields and their fluctuations. Xin et al. [69] conducted a study of a 7.1 cm methane pool fire that quantitatively reproduced the average scalars and velocities. Numerical simulations of pool fires employing the LES approach and accounting for participating media radiative heat transfer have also been demonstrated [70-72]. In the fire protec-tion engineering community, a widely used fire simulation tool is Fire Dynamics Simulator (FDS), developed by McGrattan et al. at NIST [73, 74]. This LES-based tool has been used in residential and industrial fire reconstructions and in the design of fire protection systems.

In the V&V hierarchy (Fig. 9), the low-Mach LES algorithm and the subgrid turbulence closure are identified as two of the unit problems. The LES algorithm is composed of the numerical dif-ferencing scheme and a solution method (algorithm) for solving the filtered set of governing equa-tions. The subgrid turbulence model is the set of approximations that ‘close’ the set of filtered equations, effectively modeling the unresolved turbulent fluctuations. The LES algorithm and subgrid turbulence closure are closely coupled, but by separating the two, one can independently address modeling choices that affect simulation results for the intended use of the LES tool.

6.1 LES equations

The essential governing equations, written in finite volume form, include the mass balance, momentum balance, mixture fraction balance, and energy balance equations. Using a bold-face symbol to represent a vector quantity, the equations are:

The mass balance,1.

(12)

where r is density and u is the velocity vector.

The momentum balance,2.

(13)

where t is the deviatoric stress tensor defined as 23

2 k

k

uij ij ijx

St m m d∂∂= - and the symmetric

stress tensor Sij = 1 _ 2 ( ∂ui __ ∂xj +

∂uj __ ∂xi ) . The second isotropic term in tij is absorbed into the pressure

projection for the current low-Mach scheme. Also in eqn (13), g is the gravitational body force and p is the pressure.

The mixture fraction balance,3.

(14)

where f is the mixture fraction and a Fick’s law form of the diffusion term assuming equal diffusivities results in a single diffusion coefficient, D.

The thermal energy balance,4.

(15)




where h is the sum of the chemical plus sensible enthalpy and q is the radiative flux. A Fou-rier’s law form of the conduction term is used with a diffusion coefficient, k, and the pressure term is neglected.

Now, consider a control volume, V, with surface area S. Because the equations will be solved on a computational grid, one can assume that the control volume has N faces, where unique faces are identified with their index k. The discussion is further simplified by only considering cubic volumes of length h.

Given the cubic control volume, a surface-filtered field for a variable f is defined as ___

f (j)(x), where the variable is filtered on a plane in the xj orthogonal direction. Then, for any surface k, the field is sampled at the face-centered location. For example, if j = 1, the surface-filtered quantity is

(16)

The volume average follows as

(17)

The bars over the variable f are labeled with superscripts ‘2d’ and ‘3d’ to distinguish between the two filters. Pope [75] identifies the proceeding definitions as using the ‘anisotropic box’ filter kernel where the resultant variables are simply averages over the interval 1 1

2 2j j jx h x x h- < < +′ .For convenience in isolating density in the filtered equations, a Favre-filtered quantity is

defined for an arbitrary variable j as

2d2d

d,

2

rjj

r≡

(18)

and

3d3d

3d.

rjj

r≡

(19)

This convention of explicitly defining the 2d and 3d filters is different than what is commonly observed in the literature, where the filtered equations are derived from finite difference equa-tions rather than finite volume equations. Thus, using

__ r2d and

__ r3d in eqns (18) and (19) for surface

and volume filtered densities, respectively, is appropriate for the present discussion.These definitions for filtered quantities are applied to the integral forms of the governing equa-

tions to obtain the Favre-filtered LES equations. Nevertheless, there are terms in the Favre-fil-

tered equations that cannot be solved. These include the surface-filtered convection of momentum

2d ,i ju u the surface-filtered convection of mixture fraction, 2d

,ju f and the surface-filtered convec-tion of enthalpy,

2d.ju h

For the filtered momentum product,

2d2di ju ur , a subgrid stress tensor is defined as,

sgs 2d 2d 2d.ij i ji ju uu ut = -

(20)

Similarly, subgrid diffusion terms are defined for mixture fraction and enthalpy,

= -

2d 2d 2d ,fjj u fu fJ

(21)




= - 2d 2d2d .hjj uu h hJ

(22)

Using these definitions, the final forms of the Favre-filtered equations are

The filtered mass balance,1.

3d 2d 2dd( ) ( ) 0.

dk

kj jS

n ut V

r r+ =

(23)

The filtered momentum balance,2.

sgs3d 3d 2d 2d 2d 2d 2d 3dd( ) (– )

dk

i kj i j ij ij ij iS

u n u u p gt V

r r t t d r= + + - + .

(24)

The filtered mixture fraction balance,3.

3d 3d 2d 2d 2d 2dd( ) (– ).

dfk

kj jS

f n u f D ft V

r r= + ∇ +

J

(25)

The filtered thermal energy balance,4.

r r= - + ∇ - +

2d3d 2d3d 2d 2d 2dd( ) ( ).

dhk

kj j

Sn u k h qh h

t VJ

(26)

The subgrid momentum stress, t ij sgs , the subgrid mixture fraction dissipation, ,fJ and the sub-

grid enthalpy dissipation, ,hJ contain the unresolved or subgrid action of the turbulence on the transported quantities. Since these terms arise from definitions, models are introduced to include the subgrid effects that they represent. These models are discussed next.

6.2 Subgrid turbulence models

Invoking an ‘eddy-viscosity’ modeling concept, the subgrid transport due to turbulent advection is treated as an enhanced diffusion term for the unclosed terms listed above. That is, the subgrid mixture fraction dissipation and subgrid enthalpy dissipation are respectively written as,

2df

tj

fD

x

∂= ,∂

J

(27)

and

2dh

tj

hk

x

∂= .∂

J

(28)

To model Dt and kt, constant turbulent Schmidt (Sct) number,

tt

t

1Sc

D

mr

= ,

(29)

and Prandtl (Prt) number,

tt

t

1Pr

k

mr

= ,

(30)




are assumed, where mt is a turbulent viscosity. Following Pitsch and Steiner [76], the values of the turbulent Schmidt and Prandtl numbers are taken as Sct = Prt = 0.4, which is consistent with a unity Lewis number assumption.

For the subgrid momentum stress tensor, sgs,ijt two common LES turbulence closure models are the constant coefficient Smagorinsky model [77] and the dynamic coefficient Smagorinsky model [78]. As with the scalar subgrid modeling terms, the eddy viscosity model is again invoked for

sgs,ijt which is approximated by

(31)

where Δ is the filter width, t is the eddy viscosity, and 1/ 2| | (2 ) .ij ijS S S≡ For the Smagorinsky model, Cs ≈ 2 depending on the filter type, numerical method, and flow configuration [75].

For the dynamic Smagorinsky model, Cs is computed by taking a least squares approach to determine the length scale [79],

2s( )

ij ij

ij ij

MC

M MΔ = ,

(32)

where

ij ij ijC S S C S S= Δ - Δ2 22 2( ) ( ) ,s s| | | |

(33)

and

M S S S Sij ij ij≡ 2(| | | | ). - a2

(34)

The hat defines an explicit test filter and the angular brackets in eqn (32) conceptually represent an averaging over a homogeneous region of space that, experience has shown, is necessary for stability. Experience has also shown that averaging over the test filter width is adequate. The filter width ratio, ˆ / ,= Δ Δa is usually taken to be 2.

6.3 LES algorithm

The set of filtered equations (eqns (23)-(26)) are discretized in space and time and solved on a staggered, finite volume mesh. The staggering scheme consists of four offset grids. One grid stores the scalar quantities and the remaining three grids store each component of the velocity vector. The velocity components are situated so that the center of their control volume is located on the face centers of the scalar grid in their respective direction.

The staggered arrangement is advantageous for computing low-Mach LES reacting flows. First, since a pressure projection algorithm is used, the velocities are exactly projected without interpola-tion error because the location of the pressure gradient coincides directly with the location of the velocity storage location. Second, Morinishi et al. [80] showed that kinetic energy is exactly con-served on a staggered grid when using a central differencing scheme on the convection and diffusion terms without a subgrid model. Having a spatial scheme that conserves kinetic energy is advanta-geous because it limits artificial dissipation that arises from the differencing scheme. These conser-vation properties make the staggered grid a prime choice for LES reacting flow simulations.

For the spatial discretization of the LES scalar equations, flux limiting and upwind schemes for the convection operator are used. These schemes are advantageous for ensuring that scalar




values remain bounded. For the momentum equation, a central differencing scheme for the con-vection operator is used. All diffusion terms are computed with a second-order approximation of the gradient.

When computing the 2d surface filtered field on the faces of the control volume, one is forced to use an interpolation from the 3d volume filtered field. This approximation is tolerated because computing the 2d surface field is not possible with the given grid scheme.

An explicit time stepping scheme is chosen. A general, multistep explicit update for a variable, f, may be written as,

0

1( ) ( ) ( )

0

( ) 1

,

( ( )), 1 ,

,

n

mi k k

i k i kk

m n

V t L i mb-

, ,=

+

=

= + Δ = ,...,

=

∑

f f

f a f f

f f

(35)

where n is the time level, m is the substep between n and n + 1, a and b are integration coeffi-cients, and L is a linearization operator on the convective flux and source terms.

The time step is limited by

F.E. ,t c tΔ ≤ Δ (36)

where ΔtF.E. is the forward-Euler time step limited by the Courant-Friedrichs-Levy condition and c is a constant less than or equal to 1.

A higher order multistep method is derived by letting m > 1 and choosing appropriate con-stants for a and b. For this study, two-step and three-step, strong stability preserving (SSP) coef-ficients were chosen from Gottlieb et al. [81]. The coefficients for SSP-RK 2 and SSP-RK 3 are optimal in the sense that the scheme is stable when c = 1 if the forward-Euler time step is stable for hyperbolic problems. In practice, for the Navier-Stokes equations, the value of c is taken to be less than 1.

Choosing an explicit time stepping scheme, rather than an implicit one, creates a challenge for solving the set of equations. The density at the n + 1 time step, which is required to determine the cardinal variables, requires an estimation. Taking the estimated density for

__ rn+1 to be

__ r*, the estima-

tion can be as simple as __ r* =

__ rn. Note that the 2d and 3d filter distinction is dropped for the remain-

der of this discussion for the sake of simplicity. Another procedure includes predicting a value for __ r* from performing a forward-Euler step in time as,

( ) .n nk

kj jS

t n uV

r* = r r- Δ

(37)

Ideally, one would like to know __ rn+1, but r is a function of the same variables that are being

updated in time, namely the mixture fraction, f, and enthalpy, h. This quandary, a result of the explicit time stepping method, will not be resolved for variable density flows without using a fully implicit method. Explicit methods, however, can be advantageous, especially for large scale parallel computations. Specifically, load balancing is easier and more efficient with explicit methods because the amount of work required per processor is readily determined a priori. Explicit methods are also easier to code into a computer and to debug. For these reasons, the current algorithm discussion is limited to explicit methods.

The explicit algorithm for solving the set of filtered equations is shown in Algorithm 1.




Algorithm 1 Explicit LES algorithm.for t = tmin…tmax do

for RKstep = 1…N doSolve for scalars products 1 1( ) and ( ) .n nf hr r+ +

Estimate __ r* =

__ rn+1 from eqn (37)

if __ r* <

__ rmin or

__ r* >

__ rmax then

__ r* =

__ rn

end if1 1 1 1

+1 1 1

Compute ( ) / * and ( ) / *

Compute ( , )

Compute *, the unprojected velocities

n n n n

n n n

f f h h

f f h

r r r r

r

+ + + +

+ +

= =

=u

Perform RK averaging if neededCompute correct pressure from pressure Poisson equationProject velocities with correct pressure to get 1n+u

end forend for

6.4 Large scale, parallel computing with LES

LES is computationally intensive because it resolves a relatively large set of spatial and temporal scales. An LES algorithm can be implemented in a serial code, but the underlying models must be simplified and/or lower resolution cases must be considered. To understand the interactions between a transportation fuel fire and embedded objects, all relevant scales require resolution. For example, the relevant scales for turbulence/chemistry interactions can be orders of magni-tude smaller than the largest fire scales. Accounting for all these length and time scales requires massively parallel computations.

The LES fire simulation tool described above utilizes Uintah, a component-based visual prob-lem solving environment (PSE) that provides a framework for large-scale parallelization of differ-ent applications [82-84]. Uintah was designed and implemented to satisfy three goals: (1) to provide a general framework for massively parallel simulations of fluid and particle physics; (2) to facilitate both MPI- and thread-based parallelism; and (3) to allow scientists from outside the computer field to have an intuitive method for easily inserting their algorithms into a parallel framework without being bogged down by parallel programming details.

The integration of the LES fire simulation tool in the Uintah PSE required the development of reusable, physics-based components that could be used interchangeably and interact with other components. Examples of such components include a pressure solver, a momentum solver, a scalar solver, and a subgrid scale turbulence model. Also implemented in Uintah are components developed by third parties, specifically nonlinear and linear solvers designed for complex flow problems. Realistic fire simulations must account for relevant physical processes such as turbu-lent reacting flow, convective and radiative heat transfer, multiphase interactions, and fundamen-tal gas-phase chemistry. Representations of these physical processes lead to very large sets of highly nonlinear, partial differential equations (PDEs); robust nonlinear and linear solvers on massively parallel platforms are required. Hence, two suites of nonlinear and linear scalable solvers for scientific applications modeled using PDEs, Portable Extensible Toolkit for Scientific Computation (PETSc) [85] and High Performance Preconditioners (HYPRE) [86], are interfaced with Uintah.




6.5 V&V studies of LES code/turbulence model

6.5.1 Verification using the method of manufactured solutionsBoth analytical and manufactured solutions are frequently used as verification tools. Analytical solutions to the Navier-Stokes equations usually involve simple systems where parts of the equa-tions are reasonably neglected. As a result, not all parts of the equation and the corresponding discretization scheme are fully tested when compared to analytical solutions. Manufactured solu-tions allow for arbitrary complexity in the solutions because they have no physical meaning and can be formulated to verify all parts of the governing equations. When manufactured solutions are processed through the governing equations, the governing equation itself might not be satisfied, so an extra source term is added to account for the additional terms that arise from the manufactured solution. The method of manufactured solutions [87] is an extremely useful verification exercise for finding programming errors and ensuring expected behavior of the computer code.

The convective and diffusive spatial operators as well as the pressure correction algorithm are tested in two-dimensional planes by initializing the domain with a manufactured solution for velocity and pressure (added exponential term to manufactured solution in [88]),

(38)

(39)

(40)

where A is the amplitude and is the viscosity. Note that the velocity field satisfies the continuity equation, ∇· U = 0, for constant density.

To test the spatial discretization error, the advection/diffusion terms and the computed gradi-ent of the pressure correction from the Poisson solve are evaluated at t = 0. Then, advection/dif-fusion terms and the correction gradient are compared to the exact solutions for each two-dimensional plane (x-y, x–z, y–z) in a three-dimensional Cartesian space. The total force vector on a fluid element is given by the sum of the individual components,

Total a d PF F F F∇= + + , (41)

where F a is the advective force, F d is the diffusive force, and F∇P is the pressure force. Decom-posing the force vector into its various components is useful for identifying programming error in individual force components, but here we consider only the total force vector.

The total normalized error for the force components is measured as

Total Total

TotalNormalized error

e

e

F F

F

-= ,

(42)

where the subscript e is the force computed from the manufactured solution. Figure 16 shows that the normalized error from the spatial discretization decreases at a second-order rate with increasing mesh resolution for each two-dimensional plane.

6.5.2 Verification and validation with Compte-Bellot and Corrsin dataFurther verification of the LES code and validation of the constant coefficient and dynamic Sma-gorinsky subgrid turbulence models is achieved by initializing the computational domain with the




experimental data of Compte-Bellot and Corrsin [89] and then marching the solution in time using the second-order SSP-RK time stepping scheme on a 323 periodic mesh. The curves generated by this technique are displayed in Fig. 17. The straight solid line represents a simulation with no subgrid turbulence model and no molecular viscosity. This line stays nearly level, with only a slight increase in kinetic energy that is added from the time stepping scheme (the energy characteristics of the SSP-RK algorithm are discussed in [90]). This result verifies that the simulation is free from

10−1 10010−3

10−2

10−1

100

∆x

Nor

mal

ized

err

or

x−y planex−z planey−z plane

Figure 16: Total error convergence using a manufactured solution for the spatial operators. Each two-dimensional plane in the three-dimensional Cartesian space is tested and shows second-order behavior.

Figure 17: LES code verification and turbulence subgrid model validation. Kinetic energy is reported per unit mass.




numerical dissipation. The other two curves show the kinetic energy behavior obtained from the constant coefficient Smagorinsky and dynamic coefficient Smagorinsky models. Both curves gen-erally follow the kinetic energy decay in the data, an expected result for isotropic turbulence.

6.5.3 Validation of subgrid turbulence modelsAdditional turbulence model validation is performed using buoyant helium plume data from the ‘coupled problem’ level of the V&V hierarchy. This ‘coupled problem’ combines the effects of fluid flow and turbulence without the complications introduced by chemical reactions. The data set from the 1 m helium plume, taken in the FLAME facility at Sandia National Laboratories in Albuquerque, NM, includes time-averaged vertical velocity, horizontal velocity, and mixture fraction profiles as well as instantaneous values of these variables [91].

Simulations of the 1 m helium plume were performed on a 3 m3 computational domain using the LES code described above coupled with two types of dynamic turbulence models: the dynamic coefficient Smagorinsky model described above and a local dynamic model [92]. The purpose of the study was to determine the best turbulence model for the large buoyant plume. In the case of helium, it has been observed [93] that small Rayleigh-Taylor instabilities, on the order of 1.5 cm for a 1 m helium plume, may control the strength of the air entrainment. Failure to capture this effect leads to weak air entrainment and velocities that are too high in the centerline velocity field. Since proper mixing requires that the length scale of the Rayleigh-Taylor instability be captured on the mesh, a turbulence model that does not smear out the instability is preferred.

For the simulations, the turbulent Schmidt (Sct) and Prandtl (Prt) numbers were held constant at 0.4 (eqns (29) and (30)) and the filter width (Δ) was averaged over a grid volume,

1 3( ) .x y z /Δ = Δ Δ Δ

(43)

Figure 18 compares mixture fraction as a function of radial distance for the two turbulence mod-els and three mesh resolutions at a height of 0.6 m above the inlet. The bands on the experimental data represent the 90% confidence interval constructed from the experimental data as discussed in Section 3.2. While both models overpredict the helium centerline concentration, the local dynamic model appears to perform slightly better and to converge at a lower mesh resolution than the dynamic Smagorinsky model. However, global metric values (from eqns (9) and (10)) shown in Table 4 for the mixture fraction and streamwise (u) velocity components suggest that neither turbulence model provides a distinct advantage over the other at the finest resolution (Δx = 1 cm).

Further investigation of the overprediction of the centerline helium concentration is ongoing and includes understanding the effects of the density prediction in the explicit scheme and of the scalar turbulent closure.

7 Combustion/reaction models

Detailed combustion modeling of turbulent flows is computationally prohibitive due to the wide range of time and length scales that are coupled through interactions between thermochemistry and fluid dynamics. The use of a detailed kinetic scheme to describe the chemistry requires the solution of a transport equation for NS - 1 species where NS is the total number of species. This requirement, coupled with the stiffness of the source terms in the transport equations, makes the computational load unmanageable for transportation fuel pool fires. Fortunately, the fluid dynamics length and time scales overlap with only a subset of the thermochemical time scales, so some degree of decoupling is possible. Indeed, a large class of combustion models relies on the




assumption that many chemical time scales are significantly faster than the fluid dynamic scales of interest and can be decoupled. The entire thermochemical state is then represented by a small set of parameters called reaction variables.

This system representation by a small set of reaction variables is only valid when the ther-mochemical state of the system is well-approximated by a manifold in the lower-dimensional space defined by the reaction variables [94]. The concept of a low-dimensional manifold is best explained by considering different reaction trajectories in a high-dimensional state space. Due to fast reactions, these trajectories quickly relax to a low-dimensional attracting manifold governed by the slow reactions. Once the manifold is reached, all reaction trajectories move along the manifold toward equilibrium.

The ultimate goal of a manifold identification technique is to represent the chemical and molecu-lar transport processes that control flame structure (the subgrid or microscale physics) in a mac-roscale simulation. This goal is achieved through parameterization of the state space (r, T, Y1, Y2,…,YS) described by the low-dimensional manifold. A transport equation is then solved on the computational mesh for each of the parameters. A model for all other thermochemical variables as a function of the resolved scale parameters provides the bridge between the resolved and the unre-solved scales in the simulation. This model is called a subgrid reaction model and is located at the unit problem level in Fig. 9.

Figure 18: Profiles of the average mixture fraction as a function of radial distance at a height of 0.6 m above the inlet for (a) the dynamic Smagorinsky model and (b) the local dynamic model.

(a) (b)

Table 4: Global average relative errors with the average relative confidence indicator for the streamwise (u) velocity and mixture fraction. All values are percentages.

u velocity Mixture fraction

Dynamic Dynamic Resolution Smagorinsky Local dynamic Smagorinsky Local dynamic

1123 25 ± 20 NA 91 ± 45 NA2243 18 ± 20 9 ± 20 64 ± 45 41 ± 453003 11 ± 20 10 ± 20 47 ± 45 48 ± 45




7.1 Parameterization of a reacting system

The state of a single phase reacting system with NS species requires NS + 1 variables (e.g. NS -1 mass fractions, temperature, and pressure) to uniquely specify the thermochemical state, f, of the system [95, 96]. The reaction model parameterizes f by h, a vector of parameters (reaction variables) of size n, where n < Ns + 1. The reaction model then provides a unique mapping from h to f, i.e. each fi is represented by an Nh-dimensional surface in h-space. Mathematically, the state relationship is written as

1 2 1 2( ) ( ) ( )S nT Y Y Yr h h h h, , , ,..., ≈ , ,..., =f f f (44)

Given that the thermochemical state of the physical system is inherently (NS + 1)-dimensional, a unique surface may not exist in the lower-dimensional space parameterized by h.

While parameterization of a low-dimensional manifold greatly simplifies the solution of a com-plex reacting flow by reducing the number of independent variables in the system, the choice of reaction variables is critical. The reaction variables should span both the resolved and subgrid time scales of interest and provide a reasonable representation of the subgrid scale reaction processes.

In combustion applications, mixture fraction, f, is widely used as a reaction variable. Mixture frac-tion is defined as the local ratio of the total mass originating from the fuel stream to the total mass originating from the fuel and the oxidizer streams. For describing nonpremixed systems, mixture fraction is an obvious choice for a reaction parameter since it represents the stoichiometry of the mixture. However, it does not provide any information about the intrinsic state of the system.

In the following sections, two different parameterizations are evaluated, one using DNS data [97-100] and the other using experimental data from the International Workshop on Measurement and Computation of Turbulent Nonpremixed Flames (TNF data) [101]. Both parameterizations use the concept of canonical reactors to account for the detailed chemical kinetics and subgrid transport processes.

7.2 Use of canonical reactors

The two components of a reaction model as defined here are the identification of an attract-ing manifold in thermochemical state space and the parameterization of that manifold. Three canonical reactor models are chosen for manifold extraction: an equilibrium model, a perfectly stirred reactor (PSR) model, and a steady laminar flamelet model (SLFM). Manifolds may also be extracted from other canonical reactors such as a premixed flame reactor, a laminar diffusion flame reactor, or a reactor based on the one-dimensional turbulence model of Kerstein [102], but these reactors will not be discussed further in this chapter.

The equilibrium model is based on the assumption that the chemistry is infinitely fast and hence all chemical reactions are in equilibrium. This model ignores any effects of diffusion or of transient flame behavior. The present equilibrium calculations were performed with the CAN-TERA solver [103], which uses Gibbs free energy minimization to find the equilibrium state.

The PSR model is a mathematical approximation to a well-stirred reactor. A PSR Fortran code that predicts the steady-state temperature and species com positions [104, 105] was used to gener-ate the results shown here. Since the PSR has a flow term the reaction trajectories account for chemical kinetics coupled to flow.

The SLFM model is a one-dimensional counterflow flame configuration utilizing a coordinate transformation from physical space to mixture fraction space [106]. This reaction model accounts




for stoichiometry and diffusion simultaneously, by considering a one-dimensional coordinate in the flame-normal direction. The SLFM calculations were performed with a unity Lewis number assumption.

7.3 Progress variable parameterization

The progress variable parameterization is a two-variable reaction model based on the mixture fraction and hCO2

, a progress variable derived from the CO2 mass fraction. The model is gen-erated by reparameterizing the solution to the flamelet equations by ( f, hCO2

) instead of the usual parameterization by ( f, c), where c is the scalar dissipation. The advantage of the hCO2

parameterization is that the effects of extinction may be incorporated; parameterization by ( f, c) does not capture extinction because the state variables are discontinuous with respect to c at the steady extinction limit [107]. The flamelet solutions are then tabulated as functions of ( f, hCO2

), with hCO2 defined as

2

2

COCO

Y bh

b-

= ,-a

(45)

where a = max(YCO2 | f) and b = min(YCO2

| f).

7.3.1 Generation of DNS dataDNS data for reaction model validation were obtained from a DNS code that solves the compress-ible, reacting Navier-Stokes equations using eighth-order explicit finite-differences [108] with a fourth-order Runge-Kutta method in conjunction with a temporal error controller [109]. Mixture-averaged transport is employed, with transport coefficients calculated from the Chemkin transport package [110]. Further details, including initial and boundary conditions, can be found in [107].

DNS calculations of a spatially evolving, two-dimensional, turbulent CO/H2/N2-air jet flame were used in this parameterization analysis [107]. The fuel stream composition in mole % was 45%CO, 5%H2, and 50% N2 at 300 K and the oxidizer stream was air at 300 K. These streams yield a stoichiometric mixture fraction of fst = 0.437. The kinetic mechanism employed for CO/H2 oxidation included 12 species and 33 reactions [111, 112]. The mean jet velocity was 50 m/s with a co-flow velocity of 1 m/s. The Reynolds number based on the fuel stream properties (jet width and jet velocity) was 4,600.

7.3.2 Validation of progress variable parameterizationConsider a set of reaction variables, h, used to parameterize the thermochemical state, f, of the system. One may project a DNS data set into h-space and determine a mean surface that the DNS data occupies by 〈f | h〉, the average value of the state variables conditioned on a given set of values of the reaction variables. This concept is illustrated in Fig. 19, where the data points representing realizations of the temperature (f = T) from a DNS dataset are plotted against the mixture fraction (h = f). The thick solid line represents the conditional mean of T in mixture frac-tion space, 〈T | f 〉, while the thick dashed line represents the temperature obtained if the system was in thermochemical equilibrium. The thin lines in Fig. 19 are explained below.

Given the projected data in h-space (points in Fig. 19) and the conditional mean (thick solid line in Fig. 19), the standard deviation of f from its mean in h-space is expressed as

DNS DNS 2( | ) | ,i i is h h= 〈 - 〈〉〉f f f

(46)




where fi|h represents all values of the ith state variable which correspond to the given value h, and 〈〉 indicates an average. As there may be many points in physical space that have the same h, sfi

provides a quantitative measure of the best possible performance a given model parameter-ized by h can achieve relative to the DNS data and is henceforth referred to as the ideal model performance.

The thin solid line in Fig. 19 shows sT as a function of f and provides a measure of the accu-racy with which T is parameterized by f. The data deviates from an ideal model by approximately 70 K at f = 0.43, a 4% deviation.

The dashed line in Fig. 19 represents the temperature predicted by the equilibrium model, which is a unique function of the mixture fraction for an adiabatic system. The deviation of the DNS data from the surface defined by the model may be defined as

* DNS * 2( ( )) ,i i is h h= 〈 | - 〉f f f

(47)

where f i DNS |h is a realization of the DNS data conditioned on a specific value set of h, and fi*(h)

is the ith state variable as given by the model. The thin dashed line in Fig. 19 shows the deviation, s T * , of the equilibrium-predicted temperature from the DNS data.

The actual model performance relative to the DNS data is measured by s*fi from eqn (47).

Thus, by comparing sfi and s*fi

, a quantitative measure of the performance of the given model parameterized by h is obtained.

Figure 20 shows the results of an ( f, c) parameterization of temperature for an ideal model gen-erated from the DNS data as well as the SLFM reaction model. Com paring Figs. 19 and 20, it is clear that the addition of c as a second parameter allows significantly better representation of the data than the one-parameter equilibrium model, with maximum errors of 3% and 9% for the ideal and SLFM models, respectively at fst. However, the SLFM model does deviate from the ideal ( f, c) model at both low and high c.

Figure 21 shows the results of an ( f, hCO2) parameterization of temperature for the same DNS and

SLFM reaction model data shown in Fig. 20. The progress variable parameterization of the SLFM reaction model performs nearly ideally across the entire range of hCO2

. In fact, ideal models based on

Figure 19: DNS results of CO/H2/N2-air jet flame showing temperature projected into mixture fraction space. DNS data is represented by points and the conditional mean by the thick solid line. Also shown is the equilibrium solution (thick dotted line).




an ( f, hCO2)-parameterization are consistently able to represent the state variables better than ideal

( f, c)-parameterizations for this jet flame case.

7.4 Heat loss parameterization

The heat loss parameterization is a two-variable reaction model based on the mixture fraction f, and g, a variable derived from enthalpy that represents fractional heat loss. It is defined as:

(48)

In eqn (48), ha is the adiabatic enthalpy, h is the absolute enthalpy, Tref is the reference tempera-ture, Tad is the adiabatic temperature, cp is the mixture-averaged specific heat from the adiabatic product composition, and ha, ref is the absolute enthalpy of adiabatic products at the reference

Figure 21: Parameterization of temperature by ( f, hCO2) for the CO/H2/N2-air jet flame case. Re-

sults for (a) temperature and conditional mean and (b) normalized conditional mean from an ideal model (DNS) and the SLFM model (model) are shown.

0 0.2 0.4 0.6 0.8 10

500

1000

1500

ηCO2

(a) (b)

T, σ

T (

K)

DNS

Model

σDNS

σmodel

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

0.025

ηCO2

Nor

mal

ized

σT

DNSModel

Figure 20: Parameterization of temperature by ( f, c) for the CO/H2/N2-air jet flame case. Results for (a) temperature and conditional mean and (b) normalized conditional mean from an ideal model (DNS) and the SLFM model (model) are shown.

10−1 100 101 1020

0.02

0.04

0.06

0.08

0.1

χo

Nor

mal

ized

σT

DNSmodel

0

500

1000

1500

2000

T,σ

T

DNSmodelσDNS

σmodel

10−1 100

(a) (b)

101 102

χo




temperature. The adiabatic enthalpy and temperature are the enthalpy and temperature which would exist if no energy were lost to the surroundings. The numerator is the residual enthalpy. The denominator normalizes the residual enthalpy by the sensible enthalpy of the system. When the heat loss is zero, the system is adiabatic. If heat loss is greater than zero, heat (energy) is lost from the system. If heat loss is less than zero, heat (energy) has entered the system. For unreacted fluid elements with mixture fractions near 0 or 1, the sensible enthalpy of the system is small. As a result, g can become very large near the edges of mixture fraction space.

The inclusion of heat loss accounts for changes in the enthalpy of the system due to heat transfer phenomena such as radiation. By representing enthalpy changes with g, enthalpy becomes quasi-linearly independent of mixture fraction. This representation also facilitates tabulation of reaction model results for implementation in a CFD code and allows the incorporation of local extinction in the constructed tables.

7.4.1 TNF dataDetailed measurements were taken of a methane jet [113, 114] with a fuel composition of 22.1% CH4, 33.2% H2 and 44.7% N2 by volume. The co-flow consisted of air with 0.8% H2O entering at 292 K. The stoichiometric mixture fraction was fst = 0.167. Measurements of temperature and concentrations of N2, O2, CO, H2, CO2, H2O, OH, CH4 and NO were obtained. Axial profiles (x/d = 2.5 up to x/d = 120) and radial profiles (x/d = 5, 10, 20, 40, 60, 80) of mean and rms val-ues, conditional statistics, and single shot data were taken. Typically, 800-1,000 samples were acquired at each location with uncertainties in the experimental measurements available in the listed references.

The experimental flame data was organized into bins of ( f, g). Heat loss was calculated at each data point using eqn (48). To compute the sensible enthalpy, the adiabatic composition was obtained from an adiabatic equilibrium calculation at a reference temperature of 273.15 K. Then, each data point was placed into a bin that was characterized by an ( f, g) pair of values. The valid-ity of the parameterization proposed in eqn (44) is assessed using this TNF data table.

7.4.2 Validation of heat loss parameterizationIn order to use an ( f, g) parameterization, heat loss must be present in the canonical reactor model. For the equilibrium model, heat loss was incorporated by varying the composition and enthalpy of the initial CH4/H2/N2-air mixture. For the PSR reactor, model reactor solutions were obtained for a range of mixtures (defined by the inlet equivalence ratio) at various normalized heat loss values by including heat loss from the reactor in the calculation. The volume of the reactor for the CH4/H2/N2-air case was 67.4 cm3 and the residence time was specified as 0.003 s. For the SLFM model with a unity Lewis number assumption, the adiabatic profile for enthalpy is a line connecting the enthalpy of the fuel and oxidizer streams, a direct consequence of enthalpy being a conserved scalar. To incorporate heat loss effects into the SLFM reactor model, the heat loss as defined in eqn (48) was adopted. First, the adiabatic solution was computed followed by the com-putation of the denominator in eqn (48). Next, the enthalpy profile was computed given a constant value of heat loss. The species flamelet equations were then solved, with temperature computed from enthalpy and composition computed using a one-equation Newton’s method. The maximum scalar dissipation rate was set at 20 s-1 since in buoyancy-driven flames, the scalar dissipation rate is low and does not vary much through the flow field.

The reaction model results presented here are based on the species, thermodynamics, and detailed kinetics found in the GRI3.0 scheme, but similar results could also be obtained using the surrogate JP-8 kinetic mechanism described in Section 5.




Figure 22 shows temperature and species concentrations conditioned on various values of heat loss and plotted in mixture fraction space for the CH4/H2/N2 flame. While this flame was close to adiabatic conditions, a realizable heat loss ranging from -0.02 to 0.09 was identified [115]. These plots include the experimental data along with the results from the three canonical systems described previously. Qualitatively, the temperature manifold (Fig. 22(a)) and those of the major species, including CO2 and H2O (Fig. 22(b)), are well-represented by the PSR and SLFM reaction models [115]. Reasonable predictions for some minor species such as OH, seen in Fig. 22(c), are also achieved using the nonequilibrium models. The prediction of other minor species, including NO, could be improved with the addition of a third parameter.

With this ( f,g) parameterization, qualitative analysis reveals that the nonadiabatic equilibrium calculations match the experimental data only in the lean region; significant deviations from equi-librium are noted in the near stoichiometric and rich regions of the flame. Thus, the performance of the equilibrium model relative to the TNF flame data is inferior to that of both the PSR and SLFM reaction models. Quantitative validation, although not yet completed, requires that an appropriate validation metric be applied to the results obtained from all three canonical reactors for all φi measured experimentally.

0 0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500

Mixture Fraction(a) (b)

(c) (d)

Tem

pera

ture

ExperimentEquilibriumFlameletPSR

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Mixture Fraction

Mas

sFra

ctio

n


H2O

CO2

x 10−3

Mixture Fraction

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

Mas

sFra

ctio

n


OH

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

x 10−4

Mixture Fraction

Mas

sFra

ctio

n


NO

Figure 22: Temperature and species concentrations in a CH4/H2/N2-air flame conditioned on various values of heat loss as a function of mixture fraction from both experiments and three ca-nonical reactor models: (a) temperature at g = -0.0372, (b) CO2 and H2O mass fraction at g = -0.0107, (c) OH mass fraction at g = +0.0158, and (d) NO mass fraction at g = +0.0688.




7.5 Soot models

An essential aspect of parameterization by mixture fraction in the three canonical reactor models discussed above (equilibrium, SLFM, PSR) is that all species diffuse at the same rate. However, soot is the product of a relatively slow reaction, is not in equilibrium and does not diffuse at the same rate as the molecular species. Hence, soot is not expected to correlate well with mixture fraction. Attempts to correlate the soot volume fraction with mixture fraction in calculations of turbulent diffusion flames have been carried out previously with limited suc-cess [116]. From measurements carried out in co-flow diffusion flames, Kennedy et al. [117] modeled the nucleation rate as a function of mixture fraction alone and showed the surface growth process to be the controlling mechanism in determining total soot volume fractions. A second complication presented by soot is that in strongly sooting flames, the soot can signifi-cantly alter the flame chemistry. It acts as a sink for important species such as OH and C2H2 and as a source for CO during oxidation. It also alters the heat release profile through radiative heat loss. Due to this bidirectional coupling between the soot field and the flame field, it cannot be effectively postprocessed on established flame fields as has been done with other pollutants such as NOx.

Currently, there are two approaches to modeling soot formation in a multiscale fire simulation. The first approach is to solve transport equations on the computational mesh for the variables of interest in the chosen soot model. For example, if using the Lindstedt soot model [39], transport equations need to be solved for the soot volume fraction and the soot particle number density. The second approach is to include the soot formation and oxidation processes in the subgrid scale reac-tion model, and then parameterize these slower processes with an additional ‘time’ parameter. For example, in the SLFM approach, the slow processes such as NOx or soot formation are not accu-rately captured because the flamelet equations are solved to steady state [118]. To alleviate this shortcoming, the flamelet equations can be solved in unsteady form using time as an additional parameter. The transient flamelet may be thought of as moving through the computational mesh in a Lagrangian sense. Pitsch et al. [119] linked the flamelet time to axial position in a jet based on the axial jet velocity and then performed a numerical simulation of soot formation in a turbu-lent C2H4 jet diffusion flame. In the progress variable approach, a scalar (or combination of sca-lars) that correlates monotonically with the subgrid flamelet time is employed as the ‘time’ parameter and transported on the computational mesh. This approach was first employed by Desam and Smith [120] to study NOx formation in turbulent nonpremixed jet flames.

8 Turbulence/chemistry interactions

Transportation fires are characterized by interactions between the length and time scales of the turbulent transport processes and the chemical reactions. These length and time scales may or may not overlap, as illustrated in Fig. 23. In this figure, the ‘mixing time scale’ refers to the time scales of the turbulent transport processes while the ‘chemistry time scale’ refers to the time scales of the reactions in the kinetic mechanism. The axis in Fig. 23 represents the time and length scales of the fire physics with the smallest scales on the left and the largest scales on the right. The scales resolved on the CFD mesh, the ‘macromixing’ region, represent only a small subset of the scales present in the fire. The ‘micromixing’ region is characterized by subgrid scale mixing phenomena and turbulence/chemistry interactions that are unresolved on the computa-tional mesh. The LES filter scale is the boundary between these two regions. Subgrid scale mod-els must appropriately account for these complex coupled interactions at the unresolved scale.




These subgrid interactions influence chemical source terms in scalar transport equations and the distribution of gas phase species and soot in the fire.

A mixing model (represented by the ‘subgrid mixing model’ block at the unit problems level of the V&V hierarchy) accounts for scalar micromixing, which is the subgrid variation of the scalar field from the mean scalar value transported on the mesh, by describing the statistical distribution of the subgrid scalar field.

If the joint PDF of a set of scalars j = (j1,j2,…,jn) is known, the mean value of any function of these scalars can be calculated as

(49)

where P(j1,…,jn) is the joint PDF of (j1,…,jn).Models which describe the full joint PDF of j are known as direct or transported PDF methods

[121, 122]. Direct PDF methods are often used in the simulation of turbulent flows where many chemical degrees of freedom are incorporated [123, 124], although difficulties arise in modeling the diffusion terms in the PDF transport equations. Recently, Fox and coworkers have proposed the finite-mode PDF or multi-environment PDF model [125, 126]. This model is based on dis-cretizing the joint PDF into a small number of environments or modes and then solving transport equations for the scalar concentrations in each environment along with the probability of each environment. Higher order statistics are incorporated by increasing the number of modes that are transported. In this way, joint PDFs may be discretely approximated and chemical source terms closed directly. Analogous to direct PDF methods, the primary difficulty in the multi-environ-ment PDF approach lies in modeling the diffusion between environments.

Figure 23: Length and time scales of turbulent transport processes and chemical reactions.




8.1 Validation of presumed PDF models in nonpremixed flames

An alternate approach to direct PDF methods is a class of models, presumed PDF models, where the shape of the PDF is prescribed. These models represent an approximation to eqn (49). The presumed functions are typically continuous, which implies that the presumed PDF represents all statistical moments of the variable. Presumed PDF models offer significant advantages over direct PDF meth-ods, primarily because of their relative ease of implementation into existing CFD codes. One disad-vantage is that joint composition PDFs of all the reaction model parameters are not easily presumed. As a result, statistical independence is often assumed for reaction models with several parameters,

1 1( ) ( ) ( ),n nP P Pj j j j,..., ≈ (50)

where P(j1,...,jn) is the joint PDF of (j1,...,jn) and P(j1) is the PDF of (j1). With this assumption of statistical independence, the joint PDF of the reaction model parameters is represented as a product of conditional and marginal PDFs. Then, eqn (49) becomes

(51)

Despite its limitations, this class of models is widely used. Fortunately, many reaction models currently in use have only a few parameters which are often not strongly correlated.

The issue of parameter independence in combustion systems was evaluated using TNF work-shop data for a CO/H2/N2-air flame, a CH4/H2/N2-air flame, and a piloted CH4-air flame [127]. Two models for the joint PDF of a reaction model parameterized by ( f, g) were considered. One model assumes that the parameters are independent and that the marginal PDF of heat loss is a delta function. The other model assumes that the conditional PDF of heat loss conditioned on mixture fraction is a delta function. Both models employ a marginal mixture fraction PDF.

Figure 24 shows temperature plots of the piloted CH4-air flame comparing presumed PDF model average values to experimental average values. Both PDF models use a clipped Gaussian mixture fraction PDF. For the data labeled ‘Marg. PDF’, the marginal heat loss PDF is assumed to be a delta function. For the data labeled ‘Cond. PDF’, the conditional PDF of heat loss condi-tioned on mixture fraction is a delta function. The plots include data from a third model, the mean value model, which assumes zero variance in heat loss and mixture fraction. Additional plots from the three flames for all measured species (N2, O2, CH4, CO, H2, CO2, H2O, OH, and NO) and temperature are found in [127].

Overall, the delta conditional heat loss PDF model predicts the mean scalar values better than the delta marginal heat loss PDF model, although application of an appropriate metric is needed to quantify the differences. The assumption that the conditional PDF of heat loss is a delta func-tion ensures that integration occurs over all realizable space. However, the conditional PDF model does require knowledge of the conditional expectation of heat loss. A proposed shape for this function can be found in [127]. The marginal PDF model assumes that f and g are statistically independent, resulting in integration over a constant heat loss for all mixture fractions. The experimental data is not realizable for all points in f / g space, so a normalization is performed when integrating over any nonrealizable space. This normalization prevents accurate prediction of O2 and N2.

The mean value model predictions are good only in regions far downstream in the flame where mixing of the fuel and air streams has occurred. The assumptions of the mean value model are poor in the near jet region where mixing is not complete.




8.2 Shape of presumed PDF

Two different presumed shapes for P(j) were considered for the pool fire simulations: the b-PDF [95, 128, 129] and the clipped-Gaussian PDF [107, 130, 131]. These PDFs are parameterized by the mean (

__ j ) and variance ( s j 2 ) of the variable j. Given the LES formulation of the governing

equations, variables transported on the mesh ( __

j ) are implicitly filtered. Additionally, because of the variable-density nature of the flows being simulated, the Favre-filtered form of the governing equations (Section 6) is used.

To compute PDF shape, the LES must supply both the Favre-filtered variable and its variance. A transport equation is typically evolved for the Favre-filtered variable, while the variance may

Figure 24: Temperature plots in a piloted CH4-air flame comparing presumed PDF model aver-age values to experimental average values. The plots are at (a) h/D = 7.5 cm, (b) h/D = 15 cm, (c) h/D = 30 cm, (d) h/D = 45 cm, (e) h/D = 60 cm, and (f) h/D = 75 cm, where h is the height above the burner and D is the diameter of the orifice.

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be modeled in several ways [132]. The LES algorithm described in Section 6 employs a scale similarity model [128], assuming that the small-scale statistics can be inferred from the resolved scale structures in the flow. Using the standard definition of the variance, the mixture fraction variance is modeled as

2 2 2( ),f C f fs = -

(52)

where f is the Favre-filtered mixture fraction and the coefficient C = 0.5 [133]. Lacking a true mean as required by eqn (52), the filter is used as an approximation to the mean and then multi-plied by the model parameter.

By construction, the presumed PDF for j matches the mean and variance of j. For variables which range from 0 to 1, the maximum variance is given by

2

max (1 ).js j j, = -

(53)

At maximum variance, both the b and clipped-Gaussian PDFs reduce to appropriately weighted d -functions at j = 0 and j = 1,

2 2

max( ) (1 ) ( ) ( ) ( 1) .P j jj j d j j d j s s ,= - + - , =

(54)

Likewise, at zero variance, they become a single -function at j = __

j ,

2( ) ( ) 0.P jj d j j s= - , =

(55)

Both the b -PDF and clipped-Gaussian PDF become singular at zero and at maximum variances [107], but their properties (eqns (54) and (55)) insure that the PDF does not need to be constructed or integrated at these limits. Nevertheless, inte gration of the b -PDF can be very difficult (and inac-curate) when the variance is near its maximum, even when using integration schemes designed for singular functions. The clipped-Gaussian PDF creates no integration difficulties at high variances because the singularities are treated directly with a0 and a1 [107], making the clipped-Gaussian PDF easier and computationally cheaper to integrate than the b -PDF.

9 Radiative heat transfer model

Radiation, the dominant mode of heat transfer in hydrocarbon fires, is incorporated in the V&V hierarchy at the unit problem level in Fig. 9. With the advent of massively parallel computers, per-forming realistic computations of participating media radiative transfer is increasingly tractable. In order to spatially resolve the important flow characteristics in a fire, grids containing 106-108 com-putational cells are used at every time step associated with the calculation. Parallelization of the radiation calculations by decomposing the radiation solution in spatial, angular, or energy domains is essential. A finite volume-based discrete ordinates radiation model that is decomposed in the spa-tial domain is employed. The inputs to this model are gas temperature and the concentrations of the radiatively active species (CO2, H2O, soot), which are calculated on the spatially decomposed flow grid as well as at the boundaries. The adoption of a spatial decomposition strategy for the radiation component allows easy integration with other components in the LES fire simulation tool.

9.1 Discrete ordinates method

The discrete ordinates method is based on the numerical solution of the radiative transport equa-tion (RTE) along specified directions. The total solid angle about a location is divided into a




number of ordinate directions, each assumed to have uniform intensity. Each transport equation that is solved corresponds to an ordinate direction selected from an angular quadrature set that discretizes the unit sphere and describes the variation of directional intensity throughout the domain. If zm, μm, and hm represent the direction of cosines associated with each ordinate direc-tion, k represents the absorption coefficient and Ib represents the black body emissive power, then the differential equation governing the discrete ordinates method in the absence of scattering can be written for each direction m as [134],

b

m m mm m m m

I I IkI kI

x y zz m h

∂ ∂ ∂+ + = - + .

∂ ∂ ∂ (56)

The boundary condition associated with the eqn (56), considering the surrounding surfaces to be black, is

b .mI I= (57)

If the absorption coefficient and temperature within the domain and at the boundaries are specified, eqn (56) can be iteratively solved for the directional intensities (Im) throughout the domain for each direction associated with the discrete ordinates method.

The variables of interest in most radiative transfer analyses are the distributions of radiative heat flux vectors (q(r)) and the radiative source terms (-∇· q(r)). The radiative source term describes the conservation of radiative energy within a control volume and is a source term in the total energy equation, thereby coupling radiation with the other physical processes that occur in a multi-physics application. Both of these variables are direction-integrated quantities and are read-ily determined once the distributions of directional intensities (Im) within the domain are known [135].

When using the discrete ordinates method, integrations over solid angles to obtain q(r) and -∇· q(r) are replaced by a quadrature of order n and an appropriate angular weight (wm) associated with each direction, m. The number of equations to be solved depends on the order of approxima-tion, n, used. In the work described here, n = 4 (the S4 approximation).

The discrete ordinates method is spatially decomposed to solve the RTE on parallel computers [136]. Mathematical libraries of robust, scalable, nonlinear and linear solvers developed by third parties are used to solve the matrices that result during the solution procedure [85]. The domain boundaries are assumed to be black walls at a temperature of 293 K.

9.2 Radiative properties

In order to solve for the intensities (Im) for each direction associated with the discrete ordinates method (eqn (56)), radiative properties throughout the computational domain must be specified. It is also desirable to solve the RTE in a limited number of spectral intervals or bands in the inter-est of computational efficiency. Therefore, radiative property models must be selected that are appropriate for the conditions encountered in a transportation fuel pool fire, divide the spectrum of interest into a limited number of spectral intervals and provide averaged or spectrally inte-grated radiative properties at each interval or band.

The algorithm described here requires the radiative properties in the form of an absorption coefficient. Absorption coefficients may be extracted from total or averaged transmissivity or emissivity data using Beer-Lambert’s law after specification of path length or mean beam length. However, the specification of path lengths/mean beam lengths is difficult in buoyant pool fires due to the ‘puffing’ phenomenon exhibited by such fires [137]. Also, Beer-Lambert’s




law is not valid for an absorption coefficient that has been averaged over many spectral lines. Estimating an absorption coefficient by using a single path length and Beer-Lambert’s law for the entire spatial field results in significant error in radiative field solutions [138, 139]. Never-theless, it is difficult to implement more rigorous procedures within the domain decomposition strategy employed here and therefore all absorption coefficients are computed using a single path length.

The gray model property model that has been implemented employs total emissivity data to compute absorption coefficients. The total emissivity of CO2-H2O gas mixtures is first deter-mined from a series of curve fit relations from Hottel charts for low temperature flames (300 K < T < 1,200 K), a weighted-sum-of-gray-gases model proposed by Coppalle and Vervisch [140] for high temperature flames (2,000 K < T < 3,000 K), and a linear interpolation between the two regimes at intermediate temperatures. Total absorption coefficients are then extracted from the total emissivity data after specification of a mean beam length. Details of this property model may be found in Adams [141].

The correlation of Sarofim and Hottel [63] for the emissivity of a sooting flame is employed to estimate the absorption coefficient of soot:

soot v e

c

4ln(1 350 ),k f TL

L= +

(58)

where fv is the soot volume fraction, T is the gas or soot temperature in Kelvin, and Le is the mean beam length.

To determine non-gray properties, the spectral region of interest (50 to 10,000 cm-1) is divided into a number of intervals (width ≈ 25 cm-1) and spectral optical depths are determined at each interval employing a narrow band model (RADCAL) [5]. An average absorption coefficient (kh) correspond-ing to each interval is then obtained by dividing the spectral optical depth by a path length (L). The entire spectrum is then divided into six bands and the average absorption coefficients within each band (h) are lumped together to yield a patch mean absorption coefficient for that band according to the equation

(59)

This strategy is similar to that employed by Hostikka et al. [74] for performing radiation calcula-tions in an LES fire simulation except that a Planck mean absorption coefficient was evaluated and employed in their calculations. Krishnamoorthy et al. [142] showed the advantages of employing a Patch mean absorption coefficient over a Planck mean coefficient in comparisons against non-gray benchmark problems.

The evaluation of absorption coefficients from the gray and non-gray models requires the specification of a path length or mean beam length. One-tenth of the mean beam length of the computational domain is taken as the path length by Hostikka et al. [74] in their pool fire simula-tions and is the mean beam length/path length used here.

9.3 Algorithm verification

One case used for radiation model verification is the nonhomogeneous medium benchmark introduced by Hsu and Farmer [143]. The problem consists of an isothermal unit cube with cold black walls. The interior of the cube consists of a gray, non-scattering, absorbing/emitting




material with an optical thickness (t = absorption coefficient times the side length) distribution given by

( ) 0 9 1 1 1 0 1

0 5 0 5 0 5

x y zx y zt

| | | | | | , , = . - - - + . . . .

(60)

A uniform black body emissive power of unity within the domain defines the distribution of tem-perature. Since the radiative properties, temperature, and boundary conditions for this problem are known, the RTE can be solved to determine the distributions of the radiative fluxes and the radiative flux divergence. The root mean square error norms, also known as the L2 error norms, of both radiative flux and radiative flux divergence are shown in Fig. 25.

The spherical surface symmetrical equal dividing angular quadrature scheme (SSD) [144] was employed to calculate the numerical solution accuracies plotted in Fig. 25. The results obtained by Burns and Christon [145] using the rotated LC quadrature scheme are also shown in Fig. 25

(a)

(b)

Figure 25: Numerical accuracy of quadrature schemes as a function of spatial and angular resolu-tion: (a) predicted radiative flux divergence along (x, 0, 0); (b) predicted radiative heat flux along (x, 0.5, 0).




(open symbols). The number of equations that need to be solved with the SSD1a, SSD2a, and SSD3b schemes are exactly the same as those of the rotated LC4, LC6, and LC8 quadrature sets, respectively, enabling a direct comparison of the solution accuracies of the two schemes when the same number of equations is being solved. In general, the two schemes perform equally well with error norms decreas-ing as spatial and angular resolution increases.

10 Heat transfer to an embedded object in a JP-8 pool fire

The goal of this work is to calculate the potential hazard of an explosive device immersed in a pool fire of transportation fuel. We characterize the hazard in terms of the time to ignition of the device and the violence (measured as kinetic energy of the exploded container) of the event. To accom-plish this goal, a fire simulation tool for performing scalable, parallel, three-dimensional simula-tions of a large-scale pool fire with an embedded device has been developed. This simulation tool incorporates all the fire physics components at the unit problem level of the V&V hierarchy in Fig. 9 to accurately represent the heat transfer to the device. Coupling of this fire simulation tool with an energetics material model to predict time to ignition of an explosive device is dis-cussed in Section 12.

10.1 Modified LES algorithm

The LES equations (Section 6) are modified to account for the presence of a steel-shelled container of explosive material (PBX, HMX) in the computational domain. A law of the wall approximation [146] is used for the boundary condition for the momentum transport equation. Because radiation is the dominant mode of heat transfer in heavily sooting pool fires, radiative heat transfer between the solid and the fire is modeled in the enthalpy transport equation while convection heat transfer is neglected [147]. For the solid wall boundary conditions, the wall is considered as a black body radiating at its own temperature. The solid object heats up, so the boundary condition for the fire is time varying. The turbulent conductivity is modeled in a manner similar to the turbulent diffusivity as discussed in Section 6.

The solid is modeled with the ‘material-point’ method (MPM) [148, 149], which uses material (mass) points to represent the solid and calculates stresses and heat conduction within the solid using interpolation by basis functions. The equations for the fire in the presence of an object are discretized using a finite-volume scheme, as described in Section 6. Additional details about the MPM algorithm are found in [148, 149].

10.2 Coupling between LES fire phase and container heat-up phase

Because of the wide range of time scales of the complete system (intended use) case, the simula-tion is decomposed into three distinct phases. For the first phase, the dynamic LES fire simula-tion is performed to determine a steady heat flux profile to the device. This profile is generally not symmetric and depends on such variables as the crosswind velocity, the size of the pool, and the placement of the device. This phase is characterized by simulated time scales of O(1-10 s). In the second phase of the calculation, the heat-up phase, the fire simulation is frozen. Steady heat flux values from the fire phase are applied to an MPM object representing the device embed-ded in or near the fire. As this phase develops, the steel shell and the explosive material heat up, with the two materials represented by a single temperature field. This phase, with time scales




of O(10 s-10 min), is continued until the explosive’s ignition criterion is reached. The third phase, the explosion phase, begins at the ignition point. The explosion phase is characterized by time scales of O(10-9-10-3 s) and represents the container breakup and the expulsion of the explosive.

A second simulation decomposition strategy was also tested. In this strategy, the first phase proceeds as described above. The steady heat fluxes from the fire phase are then fed to a series of one-dimensional calculations performed in the radial direction of the cylindrical object. The one-dimensional calculations compute heat transfer and pressurization along the radial direction until the ignition point of the explosive is reached, at which point the simulation terminates. This strategy does not include the details of the exploding container.

10.3 Subsystem cases: heat transfer in a large JP-8 pool fire

Data sets obtained at the subsystem level of the V&V hierarchy (Fig. 9) are limited due to harsh experimental conditions and high cost, and the errors associated with such measurements are large. Nevertheless, even limited data is useful for achieving some level of validation and error quantification, particularly since the subsystem cases include the coupling of multiple physical processes and closely mimic the intended use. Here, two experimental data sets are used in a validation exercise for the LES fire phase. These data sets include heat flux measurements made at various locations in and near large JP-8 pool fires. This validation exercise is conducted using the validation metric discussed in Section 3.2.

10.3.1 Validation data setsTwo experiments have been identified for subsystem validation purposes. The first experiment was conducted by Kramer et al. [150] at the Sandia National Laboratories Burn Site. The experi-ment was intended to measure heat fluxes from a circular JP-8 pool fire (7.16 m diameter) to a large calorimeter (4.6 m length, 1.2 m diameter, 2.54 cm wall thickness) suspended directly over the pool. After the pool was ignited, temperatures were recorded for 30 min from thermocouples fixed at various axial and azimuthal locations inside the calorimeter. From the interior thermo-couple data, heat flux measurements to the outside surface of the calorimeter were deduced using the Sandia One-Dimensional Direct and Inverse Thermal (SODDIT) code [151]. In an effort to reduce wind effects, a circular wind fence (24.4 m diameter) was constructed around the fire. Wind direction and speed were measured outside the wind fence. The average wind speed was 1 m/s with a primary direction normal to the axis of the calorimeter. Despite the wind fence, the fire was observed to lean in the primary wind direction.

The second experiment was conducted by Blanchat et al. [152] at the Sandia National Lab-oratories Burn Site to provide well-characterized environmental information relative to an open pool fire with embedded, weapon-sized calorimeters. The circular pool of JP-8 fuel mea-sured 7.9 m. Details regarding the experimental setup can be found in [152]. Four separate tests were performed on different days with different measured wind speeds and different calorimeters. Here, the focus is on Experiment #1, wherein two small calorimeters (0.3 m diameter, 0.4 m long) were positioned over the pool at radii of 1.5 m and 2.5 m and the winds were characterized as being calm (0-2.2 m/s) in a direction normal to the axis of the calorim-eters. Heat flux gauges were positioned near the ground with one gauge in the center and the remaining 48 gauges in concentric circles spaced 1 m apart along eight radial directions of the pool. No wind fence was used in this experiment; wind speeds were measured at various posi-tions around the pool.




10.3.2 Simulation details and resultsTwo JP-8 pool fire simulations were performed. In the first simulation, the large calorimeter was suspended over the pool in the same configuration as [150]. The second simulation was the same configuration as the first with the exception that it did not include the calorimeter. Both simula-tions were run on 448 processors in a 20 m × 17 m × 20 m rectangular domain with a resolution of 200 × 170 × 200. The x-axis was taken as the vertical direction. A fuel inlet with diameter 7.16 m representing the pool surface was included on the -x face while the remaining -x face was modeled with a wall boundary condition. Fuel was introduced into the domain based on a fuel regression rate of 1.6 mm/min. On the -y vertical boundary, an inlet boundary condition was used to model the crosswind and was set to 1 m/s for both simulations. The opposing verti-cal side (+y) and the top of the domain (+x) were modeled with an outlet boundary condition. The remaining vertical sides (-z and +z) had pressure boundary conditions. The entire flow was initially quiescent and fuel was introduced after the simulation began. Both simulations were run until the time-averaged heat fluxes became steady. For the first simulation, heat fluxes were extracted at different axial locations on the calorimeter surface around the azimuthal direction corresponding to the thermocouple locations in the first experiment described above. For the second simulation, heat fluxes were extracted from the pool surface corresponding to the pool surface heat flux gauges of the second experiment described above.

From the first simulation, azimuthal heat flux values at a location 1.96 m down the large calorim-eter are presented in Fig. 26. Also shown are the experimental results with 90% confidence intervals for the mean data and the estimated error. Since only one experiment was performed, the first 10 min of the data were split into three equal parts to represent three data sets. The positions of π/2 and 3π/2 correspond to the top and bottom of the calorimeter, respectively. The windward side of the calorimeter corresponds to the π position and the leeward side to the 0 position. In Fig. 26(a), the simulation data lie within the experimental confidence intervals except for the lower half of the cylinder on the windward side where the simulation underpredicts the heat flux. The size of the confidence intervals is a strong function of the wind, even with the wind fence present. That is, the heat flux is varying wildly within the first 10 min, creating a large range in which the true mean heat flux could reside. This effect is particularly noticeable in the region of highest heat flux to the calorimeter (position π/2, bottom of the device). In Fig. 26(b), the estimated error is plotted with 90% confidence intervals. Again, the error is large for the lower half of the calorimeter on the wind-ward side. However, the largest error range occurs in the region of highest heat flux to the calorim-eter (position π/2, bottom of the device).

For the second simulation, results of simulated heat fluxes to the pool surface are compared with the experimental data in Fig. 27. As with the previous data set, the temporal heat flux data were separated into four segments of equal time. The results are presented as a function of the gauge id number. Gauge #1 corresponds to the center location of the pool. Gauges #2-#9 cor-respond to the first ring and so on. Note that because the diameters of the simulated fire and the experimental fire were slightly different, the gauges corresponding to the 300 series from Blanchat et al. [152] are not included in this comparison. This second data set is better charac-terized, resulting in smaller bands for the 90% confidence interval, and most simulation data points lie within the confidence interval. Two points of higher heat flux are predicted by the simulation for Gauges #5 and #6, which correspond to the windward side of the fire. These higher simulation heat fluxes may result from holding the wind speed constant at 1m/s when the experimental wind speed varied up to 2.2 m/s. Higher wind speed results in a higher tilt to the fire, lowering heat fluxes to the windward side. From the simulation results, global metric values of the average relative error metric plus/minus the average relative confidence indicator




Figure 26: Heat flux results for simulation 1 compared to experimental data at the 1.96 m slice of the large calorimeter. (a) Experimental data with a 90% confidence interval (Exp. Mean) and simulation mean (Comp. Data). (b) Estimated simulation error with a 90% confidence interval.

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Figure 27: Heat flux results at the pool surface for simulation 2 compared to heat flux gauge data at corresponding locations. (a) Experimental data with a 90% confidence interval (Exp. Mean) and simulation mean (Comp. Data). (b) Estimated simulation error with a 90% confidence interval.

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are 11% ± 22%. In other words, the average relative error ranges from 0% to 33% with a 90% confidence.

11 Prediction of heat flux to an explosive device in a JP-8 pool fire

As stated in Section 3.1, the motivation for this work is to develop a simulation tool with the intended application of predicting heat transfer to an object in a large-scale transportation fire. While such scenarios are worth studying experimentally for hazard classification reasons, they remain expensive and dangerous to perform. Thus, a simulation tool built on a hierarchy of validation becomes one potential solution for negating the costs and risks associated with per-forming the experiment. This section focuses on the prediction of heat flux to a rocket motor in a large-scale (10-20 m) JP-8 fire for transportation hazard classification. Both the Department of Transportation (DOT) and the Department of Defense (DoD) have established testing protocols that include an external bonfire test.

The DOT external fire test calls for the explosive article to be placed on a noncombustible sur-face (steel grate) above a fuel source of wood soaked with diesel fuel or equivalent. The fire is ignited and allowed to burn for 30 min while the material is observed for evidence of detonation, explosion, etc. [153]. The DoD testing protocol requires that the test specimen be surrounded by fuel rich flames from a large open hearth containing liquid fuel such that the heat transfer to the specimen is approximately 90% radiative. Wind speeds should not exceed 5.8 m/s [154].

Simulations of a full scale bonfire test of an explosive device under wind conditions allowable under the DoD testing protocol were performed using the LES fire simulation tool described in this chapter. One objective of the simulation was to determine if higher wind speeds allowed within the DoD protocol affected the engulfment of the rocket motor in the fire, resulting in a scenario that would not qualify under the current DoD regulations of full fire engulfment.

The explosive device was represented by a 1.2 m diameter, 8 m long cylindrical steel con-tainer. The container was suspended 1 m above a 24 m × 13 m rectangular pool of JP-8 fuel. The five-component JP-8 surrogate formulation proposed by Zhang et al. [36] was used for all calcu-lations. Simulations were run at two different wind speeds, 2.2 and 5.8 m/s, the upper limit of the testing protocol.

The 5.8 m/s crosswind case was run on a 30 m × 60 m × 60 m domain with a mesh resolution of 100 × 180 × 180. The case was run on 196 processors of a massively parallel machine at Lawrence Livermore National Laboratory (LLNL). The 2.2 m/s crosswind case was run on 324 processors at LLNL on a 30 m × 30 m × 60 m domain with a mesh resolution of 150 × 150 × 220.

Volume-rendered images of the temperature field at one time slice are shown in Fig. 28 for both cases. The device is not fully engulfed in the flames in either case, but in the 5.8 m/s wind condition, the fire is blown away from the container. Figure 29 shows the volume-rendered temperature field in the 2.2 m/s crosswind case from a different angle at a later time. The region of highest heat flux to the container is at a location exposed to radiation from the leeward side of the fire.

As with the calorimeter experiments in Section 11, the wind speed significantly influences the azimuthal heat flux profile of the device. Table 5 lists mean heat fluxes obtained from the simula-tion at various locations on the surface of the device. At the lower crosswind speed (2.2 m/s), the device acts as a flame holder, leading to heat fluxes near the top of the container (position π/2)that equal or exceed those at the bottom. For the 5.8 m/s crosswind case, the heat flux at the top of the device is two orders of magnitude smaller than the heat flux at the bottom of the container. At this higher wind speed, the flame is still burning under the device but leans away from the top of the device, producing the large variation in heat flux between the top and bottom of the device.




The low heat fluxes at the top of the device are a clear indication that the container is not even partially engulfed in the fire.

Information from the LES simulation about flame location and shape, heat flux to the explo-sive device, and rate of device heat-up can be used to establish acceptable operating conditions

(a) (b)

Figure 28: Volume-rendered images of the temperature field in the JP-8 pool fire: (a) 2.2 m/s crosswind and (b) 5.8 m/s crosswind.

Figure 29: Volume-rendered image of the temperature field in the JP-8 pool fire with a 2.2 m/s crosswind. Side view showing region of highest temperature on upper leeward side of the container surface.




for the hazard classification bonfire test. At the time of this simulation, no data for this particu-lar scenario existed. Thus, the LES tool was used in a predictive manner. Error bars associated with the results from these bonfire simulations must be inferred from lower hierarchical valida-tion exercises, resulting in the qualitative statements made above regarding the effect of the wind on the flame shape and heat flux characteristics. It is recognized that for many scenarios, these types of qualitative statements are unacceptable. Indeed, in high consequence scenarios, the most valuable predictive simulation results will have quantified uncertainty. While such a simulation requirement should be considered, it is not a straightforward proposition as it involves an understanding of how errors propagate in a nonlinear fashion through the V&V hierarchy. Error quantification for multiphysics, multiscale simulations is further addressed in Section 13.

12 Predicting the potential hazard of an explosive device immersed in a JP-8 pool fire

Ultimately, we are interested in calculating the potential hazard of an explosive device engulfed in a pool fire of transportation fuel. One metric for potential hazard is the time to explosion. This section describes two methods for computing time to explosion using heat flux data from the LES fire simulation tool (see Section 10.2). The first method represents the explosive device as a three-dimensional MPM object during the heat-up and explosion phases. With this method, the large deformations caused by the device breakup are captured on the computational mesh. The second method approximates heat transfer in the explosive device with a one-dimensional model that incorporates high fidelity reaction kinetics. Both methods simulate the response of an energetic material (HMX or PBX) in a fast cook-off environment. Here, fast cook-off is defined as ignition under confinement with the energetic material exposed to high heat fluxes. Fast cook-off is a surface phenomenon. Because the thermal conductivity of HMX is very low, large temperature gradients exist within the explosive. Only a thin layer of explosive next to the inner wall of the container experiences temperature increases high enough for chemical decom-position reactions to occur. In fact, the reaction zone is likely to occur in the region where the explosive is sandwiched next to the container wall [155]. For the purposes of this section, fast cook-off occurs when the energetic material is exposed to heat fluxes in the range of 1-100 kW/m2, a typical range for transportation fuel pool fires.

12.1 Three-dimensional heat transfer, PBX combustion model

The three-dimensional heat transfer model uses the MPM [148] infrastructure as noted in Section 10.2. Because of the potentially long time to ignition, an implicit time integration

Table 5: Mean heat fluxes to the explosive device obtained from simulations at two different wind speeds.

Wind speed = 2.2 m/s Wind speed = 5.8 m/s

Axial location Position = π/2 Position = 3π/2 Position = π/2 Position = 3π/2

4 m 110.1 kW/m2 82.3 kW/m2 1.8 kW/m2 77.5 kW/m2

6 m 68.6 kW/m2 84.6 kW/m2 0.7 kW/m2 92.8 kW/m2




strategy is used to eliminate stability restrictions on the timestep [149]. A single tempera-ture field is computed for the steel and PBX, an assumption which ignores any potential gap formation due to differential thermal expansion or pressurization due to decomposition of the explosive. Heat fluxes to the container surface obtained from the fire simulation are fit to high order polynomials, which are in turn sampled at particle locations around the surface of the container and treated as source terms in the solution of the energy equations. Once the heat-up phase reaches a preset ignition temperature, the implicit MPM code transfers the data to the explicit MPMICE code [155] for the explosion; pressurization does occur in the explosion phase.

The combustion model for PBX [156] in the MPMICE code is based on a simplified two-step chemical reaction scheme introduced by Ward et al. [157] in which the solid propellant is ini-tially converted to gas phase intermediates in a thermally activated, moderately exothermic zero-order reaction; the intermediates then react to form final products in a highly exothermic, bimolecular flame reaction having zero activation energy. As the pressure increases, the increase in rate of the second reaction moves the flame closer to the propellant surface, increasing the heat feedback and the surface temperature. The increased surface temperature increases the rate of the first reaction, which further increases the rate of gas formation. The computational model implements an iterative solver that seeks a self-consistent solution to the two closed form expressions for burn rate as a function of surface temperature and surface temperature as a func-tion of burn rate and pressure.

These models for the heat transfer and explosion phases were run using heat flux data from LES simulations of a 10 cm long, 10 cm diameter steel container of PBX immersed in 0.5-1.0 m JP-8 pool fires.

12.2 One-dimensional heat transfer, fast cook-off HMX model

Heat flux data from an LES simulation of a 30 cm long, 12 cm diameter steel container immersed in a 30 cm JP-8 pool fire were extracted at 24 locations around the circumference of the steel cyl-inder. The 20 seconds of fluctuating heat flux data available from the simulation were assumed to be at quasi-steady state and were replicated to extend to the time required by the fast cook-off HMX model.

The HMX model is spatially one-dimensional, fully transient, and consists of equations for modeling the solid (condensed) phase HMX, the gas phase, and the surrounding steel container for fast cook-off conditions [158]. The steel shell provides a thermal barrier to the external heat flux. The condensed phase HMX decomposition reactions are described by distributed kinetics (calculated throughout the condensed phase, not just at the surface). The gas phase description includes a detailed chemistry model for the combustion of HMX. Solution of the PDEs results in temperature, pressure, velocity, and species mass fractions as a function of position and time. For additional details, see [158, 159].

12.3 Prediction of time to ignition and explosion violence

By coupling both the MPM/MPMICE models and the fast cook-off HMX model with the LES pool fire simulation, time to ignition for a range of conditions (labeled ‘ignition delay’ in Fig. 30) was computed using both models as shown in Fig. 30. Also included in Fig. 30 are experimental time-to-ignition data obtained by various researchers [158]. Each point for ‘Flux at steel container’ is matched with the corresponding ‘Flux at interface’ value. The heat flux at the steel/HMX (or PBX)




interface is always lower than the heat flux at the exterior of the steel container. In the limit as the heat flux at the exterior approaches zero, the heat flux at the interior will also approach zero and these two heat fluxes must converge. At the high heat flux end, the deviation between the two fluxes is large. As seen in Fig. 30, when the time to ignition is based on the ‘Flux at interface’ values, the model results fall in line with the experimental data. Alternatively, when the time to ignition is based on the ‘Flux at steel container’ values, the predicted values show a strong deviation from the experimental values. Hence, an important parameter for accurately predicting ignition delay is the flux that the explosive experiences, not the flux that the container experiences.

In addition to time to ignition data, results from the MPMICE simulations show evidence of explosion violence; data from two cases are considered here. In case 1, the 10 cm diameter con-tainer is located 0.5 m above the edge of a 0.5 m diameter JP-8 pool fire, there is no crosswind, and the fuel regression rate is 6.4 mm/min. In case 2, the container is located 0. 25 m above the edge of a 0.5 m diameter JP-8 pool fire, the crosswind speed is 4 m/s, and the fuel regression rate is 6.4 mm/min. Polynomial fits of the azimuthal heat flux data from LES pool fire simulations of the two cases are displayed in Fig. 31 for one axial location on the container. These traces are distinctly different and produce different fragmentation patterns as observed in the three-dimen-sional volume renderings of the container and propellant shown in Fig. 32. A more quantitative analysis measures explosion violence by the total kinetic energy of the exploded container. Based on such an analysis, one finds that case 1 is more violent than case 2 as seen in the kinetic energy plots of Fig. 33. Experimental results have shown that lower heat fluxes produce more violent explosions, and the simulation data in Fig. 33 mirror this observation; the heat fluxes experienced by case 2 are lower than those experienced by case 1 (see Fig. 31).

These time to ignition and violence of explosion predictions provide the perspective of overall trends in the simulation data and generally agree with available data. However, they do not achieve the desired predictivity as there are no associated error bars. In fact, it is unclear how the errors identified in previous sections of this chapter were propagated in a nonlinear fashion up through the hierarchy for this ‘complete system’ case. For this reason, error quantification and propagation (see Section 13) are essential areas of research in moving toward predictivity.

0.001

0.01

0.1

1

10

100

1000

10000

0.1 1 10 100 1000

Heat Flux (W/cm2)

Igni

tion

Del

ay (

sec)

Ali et al, 99: 0.75 atm airAli et al, 99: 1 atm airVilynov and Zarko, 89Strakovski (1989)Lengelle (1985)Atwood (1988)C-SAFE (1999)C-SAFE (2001)TangHeat Flux to HMX Surface, 1D HMX ModelLES Heat Flux to Steel Surface, 1D HMX ModelLES Heat Flux to Steel Surface, MPM ModelHeat Flux to PBX Surface, MPM Model

Flux at interface

Flux at steelcontainer

Figure 30: Ignition delay versus heat flux showing the difference between calculated interior and calculated exterior heat flux levels.




10000

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40000

30o

210o

60o

240o

90o

270o

120o

300o

150o

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Case 1Case 2

W/m2

Figure 31: Polynomial fits of the azimuthal heat fluxes at a single axial location on the steel con-tainer obtained from pool fire simulations of cases 1 and 2. Case 1 - no crosswind, con-tainer is located 0.5 m above the pool surface at the edge of fire. Case 2 - crosswind of 4 m/s, container is located 0.25 m above pool surface at fire’s edge.

(a) (b)

Figure 32: Volume rendered images of container fragmentation and propellant release from simu-lations of a 10 cm diameter steel container of PBX embedded in a 0.5 m JP-8 pool fire simulation. (a) Case 1 - no crosswind, container is located 0.5 m above the pool surface at the edge of fire. (b) Case 2 - crosswind of 4 m/s, container is located 0.25 m above pool surface at fire’s edge.




Figure 33: Total kinetic energy of all particles in the MPMICE simulation. Case 1 - no cross-wind, container is located 0.5 m above the pool surface at the edge of fire. Case 2 - crosswind of 4 m/s, container is located 0.25 m above the pool surface at fire’s edge.

2 2.5 3 3.5x 10−3

0

0.5

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2.5x 104

Time from ignition, sec

Kin

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ene

rgy,

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Case2Case1

13 Toward predictivity: error quantification and propagation

The goal of the simulation at the level of the complete system is to accurately predict heat flux to a container of energetic material immersed in a transportation fuel pool fire. Despite the methodol-ogy of a V&V hierarchy, predictivity has not yet been achieved. The validation comparisons at the subsystem involve some quantification through the use of validation metrics, while the results of the ‘complete system’ simulation are qualitative in nature and do not account for uncertainties in the experimental or the simulation data. What are needed are systematic ways to represent uncer-tainties at lower levels of the V&V hierarchy, efficient computational algorithms to propagate those uncertainties all the way up to the complete system level, methods for identifying the parameters that control uncertainty, metrics for quantifying simulation error, and datasets for validation [160]. Ultimately, the truth comes from the experimental data; it is the window on the physical world. However, in ambitious simulations of multiphysics and multiscale simulations, it is through the tight coupling of both simulation and experimental data that predictivity with uncertainty quantification will be achieved.

The field of uncertainty quantification (UQ) and error propagation in multiphysics problems is an area of active research, and it still is not clear what approach or approaches will provide the analysis tools necessary to achieve predictability. McRae [160], Marzouk and Najm [161], and Najm and coworkers [162, 163] have proposed a method for UQ based on Bayesian inference. Inferring model parameters and inputs from data is a challenging task and is known as the inverse pro blem. Marzouk and Najm have focused on using Bayesian statistics as a foundation for infer-ence [161]. Interestingly, there are strong parallels between the forward propagation of uncer-tainty and Bayesian approaches to inverse problems. Marzouk and Najm have formalized this connection and have successfully employed polynomial chaos expansion (PCE) techniques to propagate a wide range of uncertainty through the forward problem. In their approach, the model parameters and field variables are treated as stochastic quantities that can be modeled using




PCE techniques. After sampling the resultant spectral expansion, they achieve a more efficient Bayesian solution of the inverse problem [161]. A comparison of this approach to the more con-ventional UQ method of sensitivity analysis and error propagation in the context of H2-O2 igni-tion under supercritical-water conditions was performed by Najm and coworkers [163]. The results indicate that PCE methods provide first-order information similar to that from the sensitiv-ity analysis. In addition, the PCE methods preserve higher-order information that is needed for accurate UQ and for assigning confidence intervals on sensitivity coefficients. Analysis shows substantial uncertainties in the sensitivity coefficients, illustrating that these higher-order effects can be significant.

A second approach has been proposed by Frenklach et al. [164] that relies on the concept of data collaboration. Data collaboration organizes the available experimental data and its uncer-tainties together with mechanistic knowledge of the physical system using the abstraction of a dataset. A dataset unit consists of ‘the measured observation, uncertainty bounds on the measure-ment, and a model that transforms active parameter values into a prediction for the measurement’ [164]. Note that a dataset unit includes a model prediction. In its application, the concept of data collaboration recognizes that a model is only an approximation to the truth and that the truth comes from the experimentally measured data. With this dataset abstraction, numerical analysis techniques can be used to probe the dataset. For example, consistency of the model to the mea-sured data or of dataset units to each other can be determined with constrained optimization that utilizes solution mapping tools and robust control algorithms. Within the data collaboration framework, consistency thus becomes a quantifiable metric that can open up the model to a new level of interrogation such as what a low or moderate value of the metric means. Additionally, the uncertainties of the experimental data are transferred directly into the model. In one example of how to use the consistency metric, a consistency test was performed with the GRI-Mech 3.0 dataset [165], which is composed of 77 dataset units. The test identified two major outliers in the dataset. The researchers who collected the data re-examined their original observations and mod-ified the reaction times they had extrapolated, removing the inconsistency in the GRI-Mech 3.0 dataset [164]. A similar consistency analysis could be applied to the model as outliers could also indicate a problem with the model.

Neither the Bayesian inference nor the data collaboration approach has yet been applied to a complex, multiscale, multiphysics problem. However, in order to achieve predictivity, it is clear that these or other approaches must be implemented in more complex systems. The treatment of uncertainties must become more systematic. Additionally, to use either approach in problems involving heat transfer to an explosive device, a large number of dataset units need to be identi-fied and compiled in a database repository including the data sets discussed in Section 10. There is clearly much work to be done both computationally and experimentally.

14 Summary

The prediction of heat transfer to objects in transportation fuel pool fires using simulations requires the integration of complex methodologies. This chapter has summarized these meth-odologies in a manner that will assist the reader in identifying a suitable approach to this chal-lenging problem. The high cost of large-scale experiments (both real-world and simulation), combined with the greatly reduced fidelity of experimental data at this scale, provides strong motivation for the use of a computational approach that has been validated and verified in a sys-tematic manner and that includes the quantification and propagation of uncertainty from the unit problem level to the complete system level.




Acknowledgments

The authors wish to acknowledge the current and former members of their research groups whose work has been included in this chapter. Without their scholarship and hard work, this chapter would not have been possible. These individuals include Stanislav Borodai, William Ciro, Jim Guilkey, Todd Harman, Gautham Krishnamoorthy, Niveditha Krishnamoorthy, Seshadri Kumar, David Lignell, Randy McDermott, Rajesh Rawat, James Sutherland, Chuck Wight, Shihong Yan, Devin Yeates, and Hongzhi Zhang.

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chapter 3 heat transfer to objects in pool fires - wit press · chapter 3 heat transfer to objects...

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